Cambridge Primary Mathematics Workbook 4 (Cambridge Primary Maths) [2 ed.] 1108760023, 9781108760027

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CAMBRIDGE • Practice activities to help learners apply their knowledge to new contexts • Three-tiered exercises in every unit get progressively more challenging to provide all learners with appropriate points to access the topic • Varied question types keep learners interested • Write-in for ease of use • Answers for all questions can be found in the accompanying teacher’s resource

This resource is endorsed by Cambridge Assessment International Education

✓ Provides learner support as part of a set of

resources for the Cambridge Primary Mathematics curriculum framework (0096) from 2020

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✓ ✓ For Cambridge schools worldwide Developed by subject experts

Primary Mathematics Workbook 4

Workbook 4

For more information on how to access and use your digital resource, please see inside front cover.

Mathematics

9781108760027 Wood and Low Primary Maths Workbook Stage 4 CVR C M Y K

Packed with activities, including puzzles, ordering and matching, this workbook helps your students practise what they have learnt. Specific questions develop thinking and working mathematically skills. Focus, Practice and Challenge exercises provide clear progression through each topic, helping learners to start at a level that matches their confidence. Ideal for use in the classroom or for homework.

Cambridge Primary

Cambridge Primary Mathematics

Mary Wood & Emma Low

Completely Cambridge Cambridge University Press works with Cambridge Assessment International Education and experienced authors to produce high-quality endorsed textbooks and digital resources that support Cambridge teachers and encourage Cambridge learners worldwide. To find out more visit cambridge.org/ cambridge-international

Registered Cambridge International Schools benefit from high-quality programmes, assessments and a wide range of support so that teachers can effectively deliver Cambridge Primary. Visit www.cambridgeinternational.org/primary to find out more.

Second edition

Digital access

CAMBRIDGE

Primary Mathematics Workbook 4 Mary Wood & Emma Low

University Printing House, Cambridge CB2 8BS, United Kingdom One Liberty Plaza, 20th Floor, New York, NY 10006, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia 314–321, 3rd Floor, Plot 3, Splendor Forum, Jasola District Centre, New Delhi – 110025, India 79 Anson Road, #06–04/06, Singapore 079906 Cambridge University Press is part of the University of Cambridge. It furthers the University’s mission by disseminating knowledge in the pursuit of education, learning and research at the highest international levels of excellence. www.cambridge.org Information on this title: www.cambridge.org/9781108760027 © Cambridge University Press 2021 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2014 Second edition 2021 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 Printed in Dubai by Oriental Press. A catalogue record for this publication is available from the British Library ISBN 978-1-108-76002-7 Paperback with Digital Access (1 Year) Additional resources for this publication at www.cambridge.org/9781108760027 Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. Information regarding prices, travel timetables, and other factual information given in this work is correct at the time of first printing but Cambridge University Press does not guarantee the accuracy of such information thereafter. Cambridge International copyright material in this publication is reproduced under licence and remains the intellectual property of Cambridge Assessment International Education. NOTICE TO TEACHERS IN THE UK It is illegal to reproduce any part of this work in material form (including photocopying and electronic storage) except under the following circumstances: (i)

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Contents

Contents How to use this book

5

Thinking and Working Mathematically

6

1

Numbers and the number system 8

1.1 1.2 1.3

Counting and sequences More on negative numbers Understanding place value

2

Time and timetables 24

2.1 Time 2.2 Timetables and time intervals

8 14 18 24 30

3

Addition and subtraction of whole numbers 34

3.1 3.2 3.3

Using a symbol to represent a missing number or operation Addition and subtraction of whole numbers Generalising with odd and even numbers

34 39 45

4 Probability 50 4.1 Likelihood

5

Multiplication, multiples and factors 58

5.1 Tables, multiples and factors 5.2 Multiplication

6

50 58 65

2D shapes 71

6.1 2D shapes and tessellation 6.2 Symmetry

71 77

7 Fractions 84 7.1 7.2

Understanding fractions Fractions as operators

84 88

8 Angles 92 8.1 8.2 8.3

Comparing angles Acute and obtuse Estimating angles

92 97 101

3

Contents

9

Comparing, rounding and dividing 106

9.1 9.2

Rounding, ordering and comparing whole numbers Division of 2-digit numbers

106 110

10 Collecting and recording data 115 10.1 How to collect and record data

115

11 Fractions and percentages 123 11.1 Equivalence, comparing and ordering fractions 11.2 Percentages

123 129

12 Investigating 3D shapes and nets 136 12.1 The properties of 3D shapes 12.2 Nets of 3D shapes

136 141

13 Addition and subtraction 147 13.1 Adding and subtracting efficiently 13.2 Adding and subtracting fractions with the same denominator

147 152

14 Area and perimeter 157 14.1 Estimating and measuring area and perimeter 14.2 Area and perimeter of rectangles

157 165

15 Special numbers 173 15.1 Ordering and comparing numbers 15.2 Working with special numbers 15.3 Tests of divisibility

173 177 183

16 Data display and interpretation 186 16.1 Displaying and interpreting data

186

17 Multiplication and division 198 17.1 Developing written methods of multiplication 17.2 Developing written methods of division

198 204

18 Position, direction and movement 209

4

18.1 Position and movement 18.2 Reflecting 2D shapes

209 217

Acknowledgements

225

How to use this book

How to use this book This workbook provides questions for you to practise what you have learned in class. There is a unit to match each unit in your Learner’s Book. Each exercise is divided into three parts: • Focus: these questions help you to master the basics • Practice: these questions help you to become more confident in using what you have learned • Challenge: these questions will make you think very hard. Each exercise is divided into three parts. You might not need to work on all of them. Your teacher will tell you which parts to do. You will also find these features: Important words that you will use.

Step-by-step examples showing a way to solve a problem. There are often many different ways to solve a problem.

These questions will help you develop your skills of thinking and working mathematically.

5

Thinking and Working Mathematically Contents

Thinking and Working Mathematically There are some important skills that you will develop as you learn mathematics.

Specialising is when I choose an example and check to see if it satisfies or does not satisfy specific mathematical criteria. Characterising is when I identify and describe the mathematical properties of an object. Generalising is when I recognise an underlying pattern by identifying many examples that satisfy the same mathematical criteria. Classifying is when I organise objects into groups according to their mathematical properties. 6

Thinking and Working Mathematically

Critiquing is when I compare and evaluate mathematical ideas, representations or solutions to identify advantages and disadvantages. Improving is when I refine mathematical ideas or representations to develop a more effective approach or solution. Conjecturing is when I form mathematical questions or ideas.

Convincing is when I present evidence to justify or challenge a mathematical idea or solution.

7

1 Numbers and the number system 1.1 Counting and sequences Worked example 1 The numbers in this sequence increase by 30 each time. 10, 40, 70, . . . The sequence continues in the same way. Which number in the sequence is closest to 200? List the terms in the sequence. The next terms in the sequence are: 10

+30

40

+30

70

+30

100

+30

130

+30

160

+30

190

+30

200

190

220

Work out which term is closest to 200.

Answer: 190 is closest to 200.

difference linear sequence negative number non-linear sequence rule sequence  spatial pattern  square number  term 

8

term-to-term rule

220

1.1 Counting and sequences

Exercise 1.1 Focus 1 Hassan shaded in grey these numbers on a hundred square. The numbers form a pattern. 1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65

66

67

68

69

70

71

72

73

74

75

76

77

78

79

80

81

82

83

84

85

86

87

88

89

90

91

92

93

94

95

96

97

98

99 100

a What is Hassan’s rule for finding the next number?

b What is the next number in his pattern?

2 The sequence 10, 16, 22, . . . continues in the same way. Write the next two numbers in the sequence.  , 

9

1 Numbers and the number system

3 The rule for a sequence of numbers is ‘add 3’ each time.

1, 4, 7, 10, 13, . . . The sequence continues in the same way. Circle the numbers that are not in the sequence.



22  28  33  40

4 A sequence has the first term 2020 and the term-to-term rule is ‘add 11’. Write the first five terms of the sequence.  ,  

 ,  

 ,  

 ,  

5 Write the next four terms in these linear sequences. a 10, 7, 4, b −9, −7, −5, c 1095, 1060, 1025,

 ,

 ,  ,

 ,  ,

 ,

 ,  ,

 ,

Tip Remember that −9 is less than −7. –10

0 –9

–7

Practice 6 Here is part of a number sequence. The numbers increase by 25 each time.

10

25, 50, 75, 100, 125, . . . Circle all the numbers below that will be in the sequence. 355  750  835  900  995

1.1 Counting and sequences

7 Amy makes a number sequence. The first term of her sequence is 1. Her term-to-term rule is ‘add 7’. Amy says, ‘If I keep adding 7, I will reach 77.’ Is Amy correct? Explain your answer.

8 Here is part of a number sequence. The first number is missing. –5



297

–5

Tip 292

–5

287

Remember to work backwards.

Write the missing number.

9 A sequence has first term 1001 and last term 1041. The term-to-term rule is ‘add 5’. Write down all the terms in the sequence.

10 Each number in this sequence is double the previous number. Write the missing numbers.

 , 3, 6, 12, 24, 48,

Challenge 11 Write the missing number in this sequence.

1, 3, 6, 10,



Explain how you worked it out.

11

1 Numbers and the number system

12 The numbers in this sequence increase by 10 each time.

4, 14, 24, . . .



The sequence continues in the same way. Write two numbers from the sequence that make a total of 68.



and

13 Describe each of the sequences below. • Is the sequence linear or non-linear? • What is the first term? • What is the term-to-term rule? • What are the next two terms in the sequence? a 5, 9, 13, 17, . . .

b 3, 11, 18, 24, . . .

c 3, 6, 12, 24, . . .

12

Tip You might find it useful to continue writing the terms of the sequence.

1.1 Counting and sequences

14 Write a sequence containing these numbers. Your sequence must have at least one number between the two given numbers. Describe the rule you use. There could be different answers. a 1 and 10

Tip You could choose a linear or a non-linear sequence.

b 6 and 20

c 3 and 15

d 1 and 100

13

1 Numbers and the number system

1.2 More on negative numbers Worked example 2

temperature zero

Here is a temperature scale. –10

0

10

°C 20

The temperature is 1° below freezing on a cold day. Mark the position of this temperature on the scale with an arrow. Each division on the number line represents 2 units. 1° below freezing is –1° and it is half way between −2 and 0. Answer:  –10

0

10

°C 20

Exercise 1.2 Focus 1 Here is a thermometer. The arrow is pointing to 10 °C. 10°

−10

0

10

20

Draw an arrow on the thermometer pointing to −5 °C.

14

30

40

1.2 More on negative numbers

2 Here are some temperatures.

4 °C  −3 °C  5 °C  0 °C  −2 °C a Which is the warmest temperature? b Which is the coldest temperature?

3 Look at the number line.

−10 −9 −8 −7 −6 −5 −4 −3 −2 −1 0

1

2

3

4

5

6

7

8

9 10

Write where you would land on the number line after these moves.

a

c

start

count on

–4

1

start

count on

–5

3

end

start

count back

6

6

start

count back

0

9

b end d

end

end

4 Circle the larger number in each pair. Find the difference between the two numbers. Use the number line to help you.

−10 −9 −8 −7 −6 −5 −4 −3 −2 −1 0

1

2

3

4

5

6

7

8

9 10

a −6 −2 Difference: b −3 −1 Difference: c  4 −4 Difference:

15

1 Numbers and the number system

Practice 5 Here is part of a number line. Write the missing numbers in the boxes.

–10

0

10

6 The thermometer shows a temperature of –8 °C. −10



0

10

20

30

40 °C

Draw arrows on the thermometer to point to these temperatures.



−4 °C  14 °C  −1 °C

7 Write the missing numbers in these sequences. a −12, −8, b −15,

 , 0, 4, 8,  , −5, 0, 5,

 ,

8 The temperature outside when Soraya arrived at school was −1 °C. By lunchtime the temperature had risen by 8 °C. What was the temperature at lunch time?

Challenge 9 Put these numbers in order on the number line. −1  1  −2  −3  −5 0

16

1.2 More on negative numbers

10 The temperature in Amsterdam is 2 °C. The temperature in Helsinki is −7 °C. How many degrees warmer is it in Amsterdam than in Helsinki?

11 Here is a fridge freezer. The temperature in the freezer is –15 °C The temperature in the fridge is 4 °C



What is the difference in temperature between the fridge and the freezer?

12 Here is part of a number line. Write the missing numbers in the boxes.

0

100

13 Mira counts on in threes starting at −13. She says, ‘If I start at −13 and keep adding 3, I will reach 0.’ Is Mira correct? Explain your answer.

17

1 Numbers and the number system

1.3 Understanding place value Worked example 3 Which number is 10 times smaller than seven thousand and seventy? 7700  707  7007  770  7070 1000s 100s 7

10s

1s

0

7

0

7

0

7

When you divide by 10, all the digits move one place to the right.

Answer: 7070 ÷ 10 = 707

compose decompose equivalent hundred thousand million place holder regroup ten thousand thousand

Exercise 1.3 Focus 1 The distance from London in England to Budapest in Hungary is 1450 km. Write the number 1450 in words.

2 Circle the number that is five thousand and five.

18

50 005  5050  5005  50 050  5550

1.3 Understanding place value

3 The table shows the number of visitors to a sports centre during four months.



Month

Number of visitors

January

6055

February

6505

March

6500

April

6550

Which month had the most visitors?

4 Complete this decomposition.

305 469 =

+ 5000 +

+

+9

5 Heidi’s password is a 5-digit number. 1 is in the ten thousands place 2 is in the ones place 3 is in the hundreds place 4 is in the thousands place 5 is in the tens place

What is Heidi’s password? Write your answer in words and in figures.

19

1 Numbers and the number system

6 Fill in the missing numbers. 6

1400

×10

×100

÷100 ×10

32

÷10

8000

×10

÷10

÷100

×100 ×10

÷10

Practice 7 Tick the largest number that can be made using these four digit cards. 3

 

9

 

0



  Nine thousand nine hundred and three



  Nine thousand and thirty-nine



  Nine thousand nine hundred and thirty



  Nine thousand and ninety-three

 

9

8 Write in digits the number that is equivalent to 130 thousand + 3 tens.

20

÷10

1.3 Understanding place value

9 Here are four number cards.



A

eight hundred and fifty

B

five hundred and eight

C

five hundred and eighty

D

fifty eight

Write the letter of the card that is the answer to: a 85 × 10



b 5800 ÷ 10

d 58 × 10



e 580 ÷ 10



c 5800 ÷ 100 f

50 800 ÷ 100

10 Four students decompose the number 29 292. Here are the results. One answer is incorrect.



A

9000 + 90 + 20 000 + 200 + 2

B

20 000 + 9000 + 200 + 90 + 2

C

2 + 200 + 20 000 + 90 + 9000

D

2 + 200 + 20 000 + 90 + 900

Which answer is incorrect?

Challenge 11 Write in words the largest number that can be made using all the digits 3, 1, 0, 9, 7 and 5.

21

1 Numbers and the number system

12 Use the clues to solve the crossword. 1 2

3

4 5

6

Across

2. The digit in the ones place in the number 742 793.



5. Seven groups of ten.



6. The digit in the ten thousands place in 842 793. Down



1. The name for 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9.



3. The digit in the hundred thousands place in the number 814 682.



4. This digit is used to hold an empty place in a number.

13 Fill in the missing numbers. a 358 × 100 = c 29 × e

22



= 2900 ÷ 100 = 3040

b 3000 ÷ 100 = d 2700 ÷

= 27

1.3 Understanding place value

14 Here are six number cards. 10

 

100

 

1000

 

35

 

305

 

350

Use two cards to complete each calculation. You can use a card more than once.



÷

= 35



×

= 350

23

2 Time and timetables 2.1 Time Worked example 1 Tick (✓) all the digital clocks that could show the same time as the analogue clock.

11 12 1 10 2 9 3 8

7 6 5

4

  

The analogue clock shows half past two, but it could be in the middle of the night or early afternoon. Answer:

2:30 is a 12-hour digital time. 02:30 is a 24-hour digital time in the middle of the night. 14:30 is a 24-hour digital time in the afternoon. You are specialising when you choose a digital time and check to see if it satisfies the criteria that it is the same time as the analogue clock.

24

2.1 Time

a.m. analogue clock digital clock hour minute p.m. second

Exercise 2.1 Focus 1 Write the missing numbers. a 5 minutes = c 3 weeks =

b 4 hours =

seconds

d

days

e 120 seconds =

f

minutes

minutes

months = 2 years 600 minutes =

hours

2 Match each time to the correct digital clock. half past four

4 o’clock

half past three

3 Find the time intervals for each pair of dates. a

2 February 2020

23 February 2020   

b

1 January 2001

31 December 2008   

c

1 March 2009

30 November 2010   

weeks

years

months

25

2 Time and timetables

4 Circle the digital time that shows the same time as this analogue clock. 11 12 1 10 2 9 3 8



7 6 5

4

3:15     3:45     9:15     9:45

5 Complete the following table using the information given. Spoken time, 12-hour

Digital clock, 24-hour

Analogue clock

11 12 1 10 2 9 3 8

7 6 5

4

11 12 1 10 2 9 3 8

7 6 5

4

afternoon

‘eight thirty a.m.’ or ‘half past eight in the morning’

26

11 12 1 10 2 9 3 8

7 6 5

4

2.1 Time

6 Joe says, ‘To change any time after midday from 12-hour to 24-hour time you just add 12 to the minutes.’ Is Joe correct? Explain your answer.

Practice 7 Write the missing numbers. a 2 years = c

b 10 hours 30 minutes =

weeks

d

hours = 2 days 14 hours

minutes

months = 6 years

8 Complete the table to show the times shown by these clocks. Use 12-hour clock time with a.m. or p.m. 11 12 1 10 2 9 3

11 12 1 10 2 9 3 8

7 6 5

8

4

7 6 5

A

B

Clock letter

Time of day

A

evening

B

night

C

evening

D

morning

4

11 12 1 10 2 9 3 8

7 6 5

C

4

11 12 1 10 2 9 3 8

7 6 5

4

D

12-hour clock time

27

2 Time and timetables

9 Match the times to the digital clocks. Time

Digital clock

quarter past 7 in the evening twenty past ten in the morning half past two in the afternoon quarter to eleven in the morning

10 Write these times as 24-hour clock times. a 10 a.m.



b  6 p.m.

c 11 p.m.



d  8 a.m.

11 Convert the times in this sequence to 24-hour digital times. What is the next term in the sequence?

Quarter to four in the afternoon → 4.45 p.m. → 17:45 → 15 minutes to seven in the evening → ?



→

→

→

→

Challenge 12 Complete the table to show the 24-hour digital clock times. ten past four in the afternoon quarter past seven in the morning quarter to ten at night

28

2.1 Time

13 Write these times as 12-hour clock time with a.m. or p.m. a 15:10



b 23:55

c 11:10



d 03:05

14 Pierre leaves home at the time shown on this analogue clock. 11 12 1 10 2 9 3 8



7 6 5

4

He arrives at school 20 minutes later. Write the time he arrives at school in 24-hour digital time.

15 Tick (✓) the time which is closest to 3 o’clock in the afternoon.

3.35 p.m.

   13:05

   03:15

   15:25

   3.35 a.m.

16 Five girls run a race. Here are their times. Sara Milly Ingrid Petra Neve

85 seconds 1 minute 34 seconds 91 seconds 1 minute 28 seconds 100 seconds

Place the girls in order at the end of the race.

   1st

  

   2nd

   

   3rd

    

   4th

  

5th

29

2 Time and timetables

2.2 Timetables and time intervals Worked example 2

calendar

Here is part of a bus timetable.

leap year

Bergsig

12:00

14:16

14:30

16:16

time interval

Greenside

12:42

14:58

15:14

16:58

timetable

Newlands

13:22

15:35

16:00

17:36

Pablo catches the 3.14 p.m. bus at Greenside. How long does it take him to travel to Newlands? 6 mins

40 mins

Use a time line. Work out the time from 15:14 to 15:20 and then from 15:20 to 16:00.

15:14 15:20 16:00 6 + 40 = 46 minutes 60

Or, subtract 14 minutes from 60 minutes (the number of minutes in an hour)

– 14 46 Answer: It takes him 46 minutes.

You are critiquing when you identify advantages and disadvantages of each method to help you choose the best method to use.

Exercise 2.2 Focus 1 Write how many minutes are between each pair of times.

30

a

08:15

08:40

  

minutes

b

10:05

10:55

  

minutes

c

16:20

16:55

  

minutes

2.2 Timetables and time intervals

2 Write the number of minutes between each of these times: a

11 12 1 10 2 9 3 8

7 6 5

4

AM

b



11 12 1 10 2 9 3 8

  

7 6 5

4

  

minutes

  Twenty-five past eight in the morning

minutes

3 Here is a train timetable. Train timetable Train 1

Train 2

Train 3

Hightown

 9.10 a.m.

10.05 a.m.

11.00 a.m.

Newbridge

 9.25 a.m.

10.20 a.m.

11.15 a.m.

Bridgetown

 9.50 a.m.

10.45 a.m.

11.40 a.m.

Donbury

10.00 a.m.

10.55 a.m.

11.50 a.m.

a How long does it take to travel from Bridgetown to Donbury? 

minutes

b How long would you have to wait for the next train if you arrived at Hightown station at 10.30 a.m.? 

minutes

c What time is the latest train you can catch at Newbridge to arrive at Bridgetown by 11.20? 

31

2 Time and timetables

Practice 4 Write how many minutes are between each pair of times. a

11:05

11:20

  

minutes

b

13:08

13:28

  

minutes

c

14:08

14:40

  

minutes

5 a Bashir hires a bike. He must return it by 4 p.m. It is 3.25 p.m. now. How many minutes does he have left?

minutes

b Vijay hires a bike for 45 minutes. He takes the bike out at 3.10 p.m. At what time must he return the bike? 6 Alana wants to travel from Paris to London by train. She wants to arrive in London by 5.30 p.m.



Paris (depart)

12:13

13:13

14:43

15:13

16:13

London (arrive)

14:30

15:39

17:02

17:39

18:39

What is the latest time she can leave Paris?

7 Here is the morning timetable for Ollie’s class.



32

Time

Monday

Tuesday

Wednesday

Thursday

Friday

9.00–10.30

English

Maths

English

Maths

English

10.30–11.00

Break

Break

Break

Break

Break

11.00–12.00

Maths

Science

Maths

Science

Maths

What is the total number of hours spent doing Science in one week?

2.2 Timetables and time intervals

Challenge 8 Tara takes 25 minutes to walk from home to school. She arrives at school at 9.00 a.m. What time did Tara leave home?

9 This clock has been reflected in a mirror. 2 3 4



1 21 11 01 9 5 6 7

8

a What time does the clock show? b Bruno looks at the clock as he sets off walking to meet Leroy. He meets Leroy at 1.00 p.m. How long was Bruno walking? 10 Heidi goes swimming every Saturday. She goes swimming on Saturday, 1 December. Altogether, how many times does Heidi go swimming in December? 11 All buses from the bus station to the railway station take the same amount of time. Fill in the empty boxes to complete the timetable. Bus timetable Bus station

8.02 a.m.

9.05 a.m.

10.01 a.m.

Shopping centre

8.12 a.m.

9.15 a.m.

10.11 a.m.

Park

8.36 a.m.

9.39 a.m.

Railway station

8.54 a.m.

9.57 a.m.

11.03 a.m.

33

3 Addition and subtraction of whole numbers 3.1 Using a symbol to represent a missing number or operation Worked example 1

symbol

Write the missing number. 85 +

= 200

Always check whether the box represents only one digit or a complete number. You need to find the difference between 85 and 200. Method 1: Count on from 85. +15 85

+100 100

200

Method 2: Subtract 85 from 200. 200 − 85 = 115 Method 3: Use known facts. 85 + 15 = 100  so  85 + 115 = 200 Answer: 85 + 115 = 200

34

3.1 Using a symbol to represent a missing number or operation

Exercise 3.1 Focus 1 Write the missing number.

37 +

= 100

2 Write the missing number so that the scales balance. 850

150

300

3 Write the missing digits. 4



+

4

= 100

4 Write the missing number.

– 8 = 505

5 Here is a number square with two missing numbers. The numbers along each edge must add up to 80. Write the missing numbers. 30

40

10

40

20

30



35

3 Addition and subtraction of whole numbers

Practice 6 Write the missing number.

+ 7 + 8 = 28

7 Write the missing number.

− 250 = 1000

8 Write the missing number.

48 −

= 26

9 The numbers in the two circles add up to the number in the square. 5 17 12

Use the same rule to find these missing numbers. 20

63 100

36

    

10 Δ and

Δ+



Write all the possible answers for Δ and



36

are single digits =4

3.1 Using a symbol to represent a missing number or operation

11 Here are six digit cards. 1



 

2

 

3

 

4

 

5

 

6

Use four of the cards to make this calculation correct. +



= 40

Challenge 12 Complete the number sentence.

304 is

more than 296.

13 Break the 4-digit code to open the treasure chest.

65 − 58 = b



41 − 2



86 − 79 =



67 −

a

d

= 12 c

8 = 39 Tip



Code is: a

b

c

d

Write one digit in each lettered box.

37

3 Addition and subtraction of whole numbers

14 Here is a number triangle with some numbers missing. The numbers along each edge must add up to 90. Use the numbers 30, 40, 50 and 60 to complete the number triangle.

Tip You could use number counters and move them around until you find the right answer.

10

20 15 Here are five number discs. 1

38

2

3

4

5

Use each number once so the total across is the same as the total down. Find different ways.

3.2 Addition and subtraction of whole numbers

3.2 Addition and subtraction of whole numbers Worked example 2 Calculate 367 + 185. Estimate: Use any method that you feel you can use quickly and efficiently.

367 is less than 400 185 is less than 200 So 367 + 185 is less than 600 Calculate: +100 367

+

+40 467

3

6

7

1

8

5

4

0

0

300 + 100

1

4

0

60 + 80

1

2

7+5

5

2

5

+40 507

You can use jumps along a number line starting from the bigger number.

+5 547

552 Or you can set out the calculation vertically. Show as much working as you need.

Answer: 552

39

3 Addition and subtraction of whole numbers

Worked example 3 Calculate 325 − 58. Estimate: Use any method that you feel you can use quickly and efficiently.

325 is 330 to the nearest 10 58 is 60 to the nearest 10 330 − 60 = 270 Calculate: –60

265

+2

267

325

Or you can set out the calculation vertically.

325 − 58 267 200 + 110 + 15 − 200 +

50 +

8

60 +

7

Answer: 325 − 58 = 267

compose decompose difference regroup

40

You can ‘count back’ on a number line. You can count back 60 and forward 2 or count back 50 and then another 8.

You will need to decompose the hundreds and tens in 325. Show as much working as you need.

3.2 Addition and subtraction of whole numbers

Exercise 3.2 Focus 1 Complete the addition questions. +20 +3 37 + 24 =

+1

37

+40

74 + 38 =

74

–2

2 Complete the subtraction questions. –20 –5 56 – 25 =

56

–20

65 – 19 =

+1

65

41

3 Addition and subtraction of whole numbers

3 Complete the calculation 749 + 568.

749 =



700

+

+

568 =

+

+

8



+

+

17

=

4 Use the most efficient method you can to complete these ­calculations. a 102 + 48

b 154 – 140

Practice 5 The number in each brick is the sum of the numbers on the two bricks below it. 60 31 13



42

18 + 11 = 29

29 18

11

Tip You will need to use addition and subtraction to complete the walls.

3.2 Addition and subtraction of whole numbers



Complete these number walls.

13

18

23



29

17

28

19

48

37 25

31

42



6 Complete the calculation 786 + 498.

+

7

8

6

4

9

8

1

4

Add the ones

7 Use the most efficient method you can to complete these calculations. a 543 + 219

b 543 − 219

43

3 Addition and subtraction of whole numbers

8 Calculate the difference between 983 and 389.

Challenge 9 Circle three numbers that total 750.

50    150    250    350    450

10 Here are four digit cards. 2



 

4

 

6

7

 

Use all four cards to make this calculation correct. +



= 100

11 Circle the number that is closest to 900.

925    891    911    808    950

12 Write the missing digits to complete the calculations. a

1 –

3 2

44

4

b

1 –

1 4

5

5 4

3.3 Generalising with odd and even numbers

13 Naomi has six number cards. 2



 

3

 

4

 

5

 

6

 

7

She makes two 3-digit numbers and adds them together. a What is the largest total Naomi can make?



b What is the smallest total she can make?



3.3 Generalising with odd and even numbers Worked example 4 Is it always, sometimes or never true that when you add two numbers together you will get an even number? 1 + 2 = 3 which is odd

Test some examples by adding two numbers together.

2 + 4 = 6 which is even

Try to write a general statement.

Answer: It is sometimes true because when you add two numbers together the answer may be odd or even. You are generalising when you look to find a rule.

45

3 Addition and subtraction of whole numbers

counter-example even generalisation (general statement) odd

Exercise 3.3 Focus 1 Shade all the odd numbers. What is the hidden letter?

416 636



232 861 220 657 154 198 423

50

32

8

53

412

654

110

5

851 825 730

404

53

676 595 358

206

45

294 687 590

682 566 742 174 552 2 Work out these calculations:

5 + 11 =







Each one is the sum of two odd numbers. Use your answers to help you complete this general statement.

23 + 19 =

The sum of two odd numbers is always

.

3 Here are some statements about odd and even numbers. Join each calculation to the correct answer.

46



odd + odd =



even   



odd + even =



odd   



even + even =

101 + 5 =

3.3 Generalising with odd and even numbers

4 Are the following statements sometimes, always or never true? Explain each answer. a The sum of two odd numbers is even.

b The sum of three odd numbers is even.

Practice 5 Work out these calculations:

5 + 12 =





Each one is the sum of one odd number and one even number. Use your answers to help you complete this general statement.

23 + 20 =



The sum of one odd number and one even number is always

101 + 10 =

 .

6 Here are some statements about odd and even numbers. Tick(✓) the correct box next to each statement. Not true



True



odd + even = odd

  



odd + odd = even

  



odd − odd = odd

  

47

3 Addition and subtraction of whole numbers

7 Leroy says, ‘I add two odd numbers and one even number and my answer is 33.’ Explain why Leroy cannot be correct.

8 Mary says, ʻThe difference between two odd numbers is odd.ʼ

Is this always true, sometimes true or never true? Explain your answer.

Challenge 9 Work out these calculations:

5 + 11 =



213 + 35 =



Use your answers to help you complete these general statements.



The sum of two odd numbers is always



The sum of two even numbers is always



The sum of one odd number and one even number is always



22 + 19 =

432 + 79 =



34 + 56 =

876 + 432 =

 .  .

10 Here are some statements. Write true if the statement is correct. Write false if it is not correct.

48



even + even = even



odd + odd = odd



even − even = even



odd − odd = odd

 .

3.3 Generalising with odd and even numbers

11 Here are four statements about odd and even numbers. One statement is wrong. Put a cross (✗) in the box by the wrong statement.

The sum of three even numbers is 24.



The sum of three odd numbers is 22.



The sum of two odd numbers is 20.



The sum of two even numbers is 18.

12 Is it always, sometimes or never true that the sum of four even numbers will divide exactly by 4?

49

4 Probability 4.1 Likelihood Worked example 1 What is the likelihood of the spinner landing on grey? Use the language of chance.

Step 1: It is possible for the spinner to land on grey, so the likelihood cannot be described as ‘no chance’. Step 2: The spinner could also land on black or white, so the likelihood cannot be described as ‘certain’. Step 3: There are more outcomes that are grey than not grey, so it is likely the spinner will land on grey.

Check if the outcome is impossible. An impossible outcome has ‘no chance’. Check if the outcome is certain.

Are there more outcomes that are grey, or more outcomes that are not grey?

Answer: There is a good chance that the spinner will land on grey.

certain even chance good chance likely likelihood maybe no chance outcome poor chance

50

4.1 Likelihood

Exercise 4.1 Focus 1 Match the event to the description of its likelihood. The first one is done for you.

A die lands on an even number.



You will change into a fish tomorrow.

Poor chance



You will breathe today.

Even chance



You will turn left today.

Good chance



You will become famous tomorrow.

Certain

No chance

2 There are 3 balls in a bag: one white ball and two black balls. Write one of these colours in each space to make the sentences correct.



red

       

white

       



There is a good chance of taking a

ball from the bag.



There is a poor chance of taking a

ball from the bag.



It is impossible to take a

black

ball from the bag.

51

4 Probability

3 Ben used a website to simulate rolling a die 20 times. These are the outcomes.



Use a website or a real die. Roll the die 10 times. Draw the outcomes here.



Complete this table to show the total rolls for you and Ben. Number

Total

1 2 3 4 5 6

52



Complete the sentence:



There is no chance of rolling a number

 .

4.1 Likelihood

Practice 4 Malik takes a sock without looking. Circle whether the sentences are true or false.

a It is certain that the sock will be striped.

True / False

b There is no chance of taking a plain sock.

True / False

c There is a good chance of taking a sock that is not spotty.

True / False

d There is a poor chance of taking a sock that is not checkered.

True / False

e There is an even chance of taking a striped sock.

True / False

f

Maybe Malik will take a spotty sock. 

True / False



Write a true sentence of your own about taking a sock without looking. Use the language of chance.

5 Taking the ball with the star wins the game. Which bag has the greatest chance of winning? A

 B

 C

 D

53

4 Probability

6 Rose used an online random number generator to pick 20 numbers from 1 to 10. These are the outcomes.



4

6

4

10

3

3

8

1

7

9

4

2

10

9

5

2

1

10

3

6

Use a random number generator to choose 10 more numbers. (If you cannot use an online random number generator you could make number cards 1 to 10 and pick one without looking 10 times.)



 

 



 

 





 

 

Complete the table to show the outcomes for you and Rose. 1

11

21

2

12

22

3

13

23

4

14

24

5

15

25

6

16

26

7

17

27

8

18

28

9

19

29

10

20

30

Complete the sentences. Use words from the vocabulary box at the start of this unit to help you. a There is

of getting 11.

b It is certain to be a number less than

54

 

c There is a

of getting a 2.

 .

 

 

4.1 Likelihood

Challenge 7 Malik takes a T-shirt from this rail without looking. Write sentences using the language of chance about which colour T-shirt will be taken from the rail.

a Use no chance in your sentence.

b Use certain in your sentence.

c Use poor chance in your sentence.

d Use likely in your sentence.

e Use even chance in your sentence.

55

4 Probability

8 Taking the ball with the star wins the game. Put these bags in order from the greatest to the least chance of winning. A

B

D

E

C



Greatest chance Least chance



E





A

9 Look at the words and phrases you have been using to describe likelihood in this Workbook and in your Learner’s Book. Write the words and phrases you feel confident using in this box.





56

Write the words and phrases that you need to improve your use of in this box.

4.1 Likelihood



Draw some different-coloured balls in this bag.



Write sentences about the likelihood of taking different-coloured balls from the bag to practise using the words and phrases you need to improve.



Ask someone to check your use of the words and phrases.

57

5 Multiplication, multiples and factors 5.1 Tables, multiples and factors Worked example 1 Write the missing factors of 20.  ,

Factors of 20 = 1,

 ,

 ,

 , 20

Here is a factor bug. 1

20

2

10

4

20

5

It is important to find all the factors of a number, so you need to be systematic. You can write the factors of 20 on the factor bug’s legs:

Tip Instead of a factor bug you could draw different rectangles using 20 squares. The factors are the lengths and widths.

• Start with 1. 1 × 20 = 20 • Try 2.

2 × 10 = 20

• Try 3.

20 ÷ 3 leaves a remainder so 3 is not a factor of 20

• Try 4.

4 × 5 = 20

• There are no more numbers to try as you have already included 5. Answer: Factors of 20 = 1, 2, 4, 5, 10, 20

array factor inverse operations multiple product

58

5.1 Tables, multiples and factors

Exercise 5.1 Focus 1 On the hundred square: • colour all the multiples of 2 in one colour • colour all the multiples of 5 in a different colour. • colour all the multiples of 7 in a different colour.



What do you notice about the multiples of 10? 1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65

66

67

68

69

70

71

72

73

74

75

76

77

78

79

80

81

82

83

84

85

86

87

88

89

90

91

92

93

94

95

96

97

98

99 100

2 Write the missing numbers in this multiplication grid.



×

3

2

6

8

4

12

16

18

5 10

30

59

5 Multiplication, multiples and factors

3 Complete this cross number puzzle. 1

2

4

3

5

6

7

8



9

ACROSS

DOWN

1. 6 × 8 2. 9 × 9 4. 24 ÷ 6 5. 63 ÷ 7 8. 10 × 6 9. 7 × 2

1. 3. 6. 7.

4 The factor pairs of 8 are:

1    and    8



2    and    4



60

Write all the factor pairs of 18.



1    and   



2    and   



   and   

6×7 3×6 6×6 8×3

5.1 Tables, multiples and factors

5 Complete the factor bugs for 36 and 45. 1

36 36

45





Practice 6 The number in each brick is the product of the two numbers below it. 6 2





3

Write the missing numbers in these diagrams.

5

3

2



6

5

7

42



7

9

7 Use these signs:

4 =

<



6

8

>

Write the correct sign in each box. a 3 × 8 

  5 × 5

b 6 × 4 

  4×6

c 7 × 8 

  6 × 9

d 4 × 4 

  2 × 8

61

5 Multiplication, multiples and factors

8 Circle all the numbers that are not multiples of 7.

7

17

27

37

47

57

67

77

87

97

9 Shade any multiples of 7 on this grid.



37

38

39

47

48

49

57

58

59

10 Here is a hexagon maze.

You need to go from the centre to one of the outside hexagons in two steps. 1 5 25

11 2

10 16

6 13

Start 9

4

17 14

18 20

3 15

7

• Start in the centre. • The next hexagon must be a multiple of 2. • The next hexagon must be a multiple of 5.

What are all the possible paths you could take?

11 Circle all the factors of 12. 1

62

2

3

4

6

8

12

24

36

72

5.1 Tables, multiples and factors

Challenge 12 Complete these multiplication triangles.

The product of the two circles on each line is the number in the square. 5

6

5

4

24

7



9

28

7

20

21

4



9

13 Complete the multiplication grids. ×

3

7

9

4

2

×

5

3

10

9 12

25 49

6 6

16 8



36 100

63

5 Multiplication, multiples and factors

14 Here are four digit cards.



2  

3  

4  

9

Use each card once to make a total that is a multiple of 7. +

15 Cross out two numbers so that the sum of the remaining numbers in each row and column is a multiple of 5.



1

2

4

8

5

6

2

3

7

7

1

3

2

6

3

9

16 Saira is thinking of two different numbers.

They are both factors of 12. The difference between the two numbers is 4.

64

What are Saira’s numbers?

5.2 Multiplication

5.2 Multiplication Worked example 2

associative law carry

Calculate 345 × 9. Estimate: The answer will be less than 350 × 10 = 3500 Grid method: ×

300

40

9

2700 360

Decompose 345 into hundreds, tens and ones.

5

Multiply 300 by 9, 40 by 9 and 5 by 9.

45

Add the products to give the answer.

2700 + 360 + 45 = 3105 Answer: 3105

Estimate first by rounding 345 to 350 and 9 to 10.

Tip You can use the same methods for multiplying 3-digit numbers by 1-digit numbers as you did for multiplying 2-digit numbers by 1-digit numbers.

Exercise 5.2 Focus 1 Complete this calculation. 6 × 15 = 6 × 5 × 3 =

×3

= 90

65

5 Multiplication, multiples and factors

2 The numbers in the circles are multiplied together to give the numbers in the squares between them. Fill in the missing numbers. 13

34

3

5

2



7

14

5

3

4

21



3

3 What is double 78?

4 Find the product of 58 and 9.

5 Here are some number cards.

66

1  

2  

3  

12  

18  

36

Use each card once to make three products with the same answer.



×

=



×

=



×

=

5.2 Multiplication

Practice 6 a Emma and Astrid calculate 9 × 2 × 5.



Complete their calculations. Who chose the better method? Emma’s method

Astrid’s method

9×2×5

=

×

=



9×2×5

=

×

=

b Mario and Ian calculate 2 × 5 × 7.

Complete their calculations. Who chose the better method?

Mario’s method

Ian’s method 2×5×7

2×5×7

=



=

×

=

×

=

67

5 Multiplication, multiples and factors

7 Work out these calculations. Show your methods. Remember to estimate first. a 25 × 8

b 69 × 6

c 76 × 9







8 Circle all the products equal to 2400.

900 × 3     300 × 8     600 × 4



300 × 7     400 × 6     800 × 3

9 Pierre uses the grid method to work out his calculations, but then spills ink on his work. What numbers are under the ink blots?



147 × 3 = ×

100

40

7

3

300

21

21

=

10 Write what the missing digits could be.



68

×

= 750

How many different answers can you find?

5.2 Multiplication

Challenge 11 Amy and Heidi work out 6 × 15. Amy’s method

Heidi’s method

6 × 15 = 6 × 5 × 3

6 × 15 = 3 × 2 × 15

= 30 × 3 = 90

= 3 × 30 = 90



Which method do you like best?



Explain why.



Write two other methods for calculating 6 × 15.

12 Estimate the following first, then calculate. Show your working. a 318 × 2

b 426 × 3

c 512 × 7







13 Adah is thinking of a number. She divides the number by 3 and her answer is 234.

What number is Adah thinking of?



69

5 Multiplication, multiples and factors

14 Write the same digit in each box to make the calculation correct. 3

6

×



3

5

6

4

15 Use the digits 3, 6, 7 and 8 to make the largest product.



70

×

=

6

2D shapes

6.1 2D shapes and tessellation Worked example 1 Tessellate this shape.

The shape has 6 sides and 6 vertices. It is a hexagon. Make a template of the shape by tracing it onto card. Draw around the template. Move the template so that the edge lines up with the shape you have already drawn. Draw around the template. Continue until you have filled the space with the tessellating shape. The hexagon tessellates on its own without any gaps.

2D shape parallel polygon regular tessellation

71

6 2D shapes

Exercise 6.1 Focus 1 Find your way through the polygon maze. Only travel through squares marked with a polygon. Only travel vertically or horizontally between squares. You could use the polygon checklist to help you find the polygons. Enter

Tip Polygon checklist A polygon: • is a 2D shape • is a closed shape • has only straight sides and no curved sides.

72

6.1 2D shapes and tessellation

2 Colour the pieces of the tangram you could use to make each shape. a

b

c

3 Trace this regular triangle and make a template of it from card.



Use your template to tessellate the triangle in the space below.



How many triangles fit in the space?

73

6 2D shapes

Practice 4 Colour the pieces of the tangram you could use to make each shape. a



Tip

Name the shapes used to make the new shape.      



Describe the new shape.



The new shape

b



Name the shapes used to make the new shape.    

   

   

74



Describe the new shape. Include the number of right angles and pairs of parallel sides the shape has.



The new shape

The shape is a different way round from how it appears in the tangram. You might want to use tracing paper to copy the shape and lay it on the tangram.

6.1 2D shapes and tessellation

c



Name the shapes used to make the new shape.   

  



Describe the new shape. Include the number of right angles and pairs of parallel sides the shape has.



The new shape

5 Trace the two shapes in this tessellating pattern. Use your tracing and a pencil and ruler to continue the pattern to fill the space below.



What two shapes make the tessellating pattern?

75

6 2D shapes

Challenge 6 How can the pieces of the tangram be rearranged so that they fit exactly into the grey shape? Draw lines on the grey shape to show how the pieces fit.



Write four things to describe the properties of the grey shape.



The grey shape:

Tip You could trace and cut out the pieces.

• • • • 7 Trace the shapes in this tessellating pattern. Use your tracing and a ruler and pencil to continue the pattern to fill the space below.

76

6.2 Symmetry

6.2 Symmetry

horizontal  line of symmetry symmetry vertical

Worked example 2 Find all of the lines of symmetry in this pattern. How many lines of symmetry does the pattern have?

Put your mirror onto the pattern. The base of the mirror must go through the middle of the pattern.

Move the mirror around the shape until the reflection shows exactly the same pattern as is on the shape behind the mirror. Try placing the mirror along the possible vertical and horizontal lines of symmetry first. Then try diagonal lines.  

When you find a line of symmetry, remove the mirror and draw the line onto the pattern. Continue moving the mirror and checking for lines of symmetry until you have found all of them. Count the number of lines.

Answer: The pattern has 4 lines of symmetry.

77

6 2D shapes

Exercise 6.2 Focus 1 Use a mirror to find all of the lines of symmetry on this pattern. Draw the lines of symmetry onto the paper with a ruler.

2 Each of these shapes has one line of symmetry. Find and draw the line of symmetry onto the shapes.

78

6.2 Symmetry

3 This is a square. It has four lines of symmetry. Draw all of the lines of symmetry onto the square.

Practice 4 Draw all of the lines of symmetry on this square mosaic design.



How many lines of symmetry does it have?

79

6 2D shapes

5 Copy and complete the table for the octagons. A

B

Octagon

C

Number of lines of symmetry

A B C D 6 Lines of symmetry have been drawn on this shape. One of the lines is wrong. Circle the line that is not a line of symmetry.

80

D

6.2 Symmetry

Challenge 7



This square mosaic design has a line of horizontal symmetry and a line of vertical symmetry. a Draw any other lines of symmetry on the mosaic that you can find. b How many lines of symmetry does the mosaic pattern have?

c If a pattern has a horizontal line of symmetry and a vertical line of symmetry, will it always also have a diagonal line of symmetry?

Explain your reasoning.

81

6 2D shapes

8 This rectangular mosaic design has a line of horizontal symmetry and a line of vertical symmetry.

a Draw any other lines of symmetry on the mosaic that you can see. b How many lines of symmetry does the mosaic pattern have?

9 a Draw any lines of symmetry on the mosaic that you can see.

b How many lines of symmetry does the mosaic pattern have?

82

6.2 Symmetry

10 Draw all of the lines of symmetry on these regular polygons. a

b



Name of shape:



Name of shape:



Number of lines of symmetry:



Number of lines of symmetry:

c



Name of shape:



Number of lines of symmetry:

83

7 Fractions 7.1 Understanding fractions Worked example 1 1 ​​  shaded. Tick (✓) all the fractions that have ​​ _ 4 A

B

C

D

Answer: A

B

A  ✓  1 part out of 4 equal parts is shaded. B  ✗  The 4 parts are not equal. C  ✓  The 2 shaded parts are equal to 1 large rectangle.

C

D

denominator fraction numerator

84

D  ✓  The shaded square is one quarter of the total area.

7.1 Understanding fractions

Exercise 7.1 Focus

1 Tick (✓) the shapes which have _ ​​ 1 ​​  shaded. 2

2 Here is part of a number line. Write the missing fraction.

0

1 4

1

1 2

3 The diagrams show fractions with a numerator of 2. The denominators are different.

2 6

2 3

2 4



Write fractions to complete the number sentences.



_ ​​  2 ​​  is greater than



Write the fractions in order, starting with the smallest.

4

2 ​​  is less than _        ​​  4

85

7 Fractions

Practice 4 Which shapes have been divided into quarters?

A

B

C

D

5 Zina has a jug of water.

E

1 litre

a What fraction of a litre of water is in the jug?

1 2

b How much water does Zina need to add so that the jug contains 1 litre?

1 4

6 Look at the number wall. It is not complete. 1 1 2 1 4

86



Use it to help you complete the number sentences. Use the signs > or < .



_ ​​  1 ​​  

4

1 ​​    ​​ _ 6

1  ​​    ​​ _ 12

2  ​​    ​​ _ 12

2 ​​    ​​ _ 6

2 ​​    ​​ _ 3

2  ​​   ​​ _ 12

7.1 Understanding fractions

Challenge 7 Shade the given fraction of each of these shapes. A 3 4

B 4 6

D 3 8

E 9 14 G 4 6

F 2 3



C 2 5

H 8 12

What do you notice about the fractions of F, G and H?

8 Here is a number line. B



0

A

1



Write the value of A as a fraction.



Write the value of B as a fraction.

87

7 Fractions

9 Is the following statement always, sometimes or never true? If you divide a shape into four parts you have split it into quarters. Explain your answer.

7.2 Fractions as operators Worked example 2 There are 40 pages in Bruno’s book.

operator unit fraction

He reads _ ​​  1 ​​  of the book on Monday. 5 How many pages does he have left to read? _ ​​  1 ​​  of 40 = 40 ÷ 5

5 40 ÷ 5 = 8

Work out how many pages Bruno reads on Monday.

So Bruno has read 8 pages. 40 − 8 = 32 Answer: Bruno has 32 pages left to read.

88

Subtract to work out how many pages are left.

7.2 Fractions as operators

Exercise 7.2 Focus

1 Here are 15 counters. Draw a ring around _ ​​ 1 ​​  of them. 5

2 What is one sixth of 24?

3 Draw a line from each calculation to the correct box.

_ ​​  1 ​​  of 20

Answer less than 10



_ ​​  1 ​​  of 60

Answer equal to 10



_ ​​  1 ​​  of 32

Answer more than 10



_ ​​  1 ​​  of 30

2 5 4 3

4 Leo takes $40 on a shopping trip.

He spends _ ​​  1 ​​  of his money. 5



How much money does he spend?

5 Use this strip to help you find _ ​​ 1 ​​  of 56. You can use the diagram to help you. 7 56

1 of 56 = 56 ÷ 7

=

89

7 Fractions

Practice 6 Jamil has 16 cards. He gives a quarter of his cards to his friend. How many cards has Jamil got left?

7 Find _ ​​  1 ​​  of 21 cm 3

8 Each number in this sequence is a quarter of the number before. 512

128

32

Write the number in the final box.

9 Find the odd one out.

1 ​​  of 8 ​​ _ 1 ​​  of 15 ​​ _ 1 ​​  of 12 _ ​​  1 ​​  of 6 ​​ _



Show working to explain your answer.

3

4

5

6

10 Divide the rectangle into four parts.

90



1 ​​ , ​​ _ 1 ​​  and ​​ _ 1  ​​ of the rectangle. The pieces must be _ ​​ 1 ​​ , ​​ _ 2 4 6 12



They must not overlap.

7.2 Fractions as operators

Challenge 11 Parveen has a packet of 20 balloons.

_ ​​  1 ​​  of the balloons are red.



How many balloons are not red?

4

12 Nasreen has a packet of coloured beads. The packet contains 5 orange beads, 5 red beads and 10 blue beads. Nasreen says, ‘Half the beads are blue.’ She is correct. Explain how you know.

13 Look at this pattern. 1 of 4 = 1 4

1 of 8 = 2 4

1 of 12 = 3 4

Write the next three numbers in the sequence.

14 What is the missing number?

1 ​​  of _ ​​  1 ​​  of 30 = ​​ _ 3

2

1 ​​  of $60 because _ 1 ​​ .’ 15 Hassan says, ‘I would rather have _ ​​ 1 ​​  of $36 than ​​ _ ​​  1 ​​  is bigger than ​​ _ 4 4 3 3 Do you agree with Hassan? Explain your answer.

91

8

Angles

8.1 Comparing angles Worked example 1 Which of these angles is smaller?

A

B Use tracing paper and a ruler. Trace one of the angles with the tracing paper and a ruler.

A

B Place the traced angle over the other angle to see which angle is greater. Tip

A

B Match one line and the points of the angles

Answer: Angle A is smaller than angle B.

92

Notice that the length of the lines and the thickness of the lines do not change the angle.

8.1 Comparing angles

angle compare degrees

Exercise 8.1 Focus 1 Circle the angle that is greater in each pair. a

b

B

A

A

B

d

c A B

A e

B

f A A

B

B

2 These angles are in order from smallest to greatest. One ­angle is in the wrong place. Circle the angle in the wrong place. a

A

B

C

D

E

93

8 Angles

b

A

B

C

D

E

c

A

B

C

D

E

Practice 3 Use tracing paper to find out which angle is smaller. Circle the smaller angle. a

b

A

B B

A

c d

A B

B

A

f e A

B A

B

4 Which pair of angles was the most difficult to compare in question 3? Explain why.

94

8.1 Comparing angles

5 Use a ruler to draw an angle that is greater than the first angle and smaller than the last angle. a

b

c

Challenge 6 Put these angles in order from smallest to greatest. A

B

C

D

E

95

8 Angles

7 Put these angles in order from smallest to greatest. E

A

D

B

C

8 Explain in your own words how to compare the sizes of two angles.

96

8.2 Acute and obtuse

8.2 Acute and obtuse Worked example 2 Is this angle acute or obtuse?

An acute angle is less than 90 degrees, it is less than a right angle or quarter turn. Compare the angle to a right angle.

The angle is greater than a right angle so it cannot be acute. 0 degrees

Two right angles 180 degrees

The angle is less than two right angles, or half turn.

Compare the angle to two right angles. An obtuse angle is: •  greater than a right angle •  less than two right angles.

Answer: The angle is obtuse.

acute angle  obtuse angle  right angle

97

8 Angles

Exercise 8.2 Focus 1 Find the angle words in the word search. f

p m e

o

b

t

u

s

e

acute

w z

q

s m a

l

l

e

r

angle

t

c

t

o

a

c

u

t

e

compare

s w o

i

f

r

a w d

a

degrees

u

e m m e

i

c

k

v

v

estimate

d

x

p

a

q

g

c

v

z

n

greater

z

a

a

t

k

h

a

j

v

h

obtuse

k

g

r

e

a

t

e

r

h

g

g

d

e

g

r

e

e

s

i

g

v

j

a

n

g

l

e

b

h

n

v

right smaller

2 Circle all of the acute angles. a

e

b

c

d

f g

3 This shape has some acute angle corners and some obtuse angle corners. Write obtuse or acute to label the other angles of the shape. Obtuse

98

8.2 Acute and obtuse

Practice 4 Use a ruler to draw five angles in each box that match the heading. Acute angles

Obtuse angles

5 The numbers in this grid are all degrees of turn between 0 and 180. Colour a path from the ‘start’ to the ‘end’ of the maze. Only pass through squares with a number of degrees that is an obtuse angle. Only move vertically or horizontally. start 120

45

60

10

50

80

100

130

165

100

175

70

20

150

165

70

35

70

130

40

160

110

10

60

65

80

145

20

70

50

60

30

20

150

160

70

150

120

100

40

50

100

60

20

110

40

160

80

10

170

30

40

135

70

110

145

30

155

125

105

170

60

30

end 120

99

8 Angles

Challenge 6 a Draw a triangle that has three acute angles.

b Draw a triangle that has a right angle. c Draw a triangle that has an obtuse angle.

d Label each angle in your triangles as ‘acute’, ‘obtuse’ or ‘right angle’. 7 Write ‘obtuse’ or ‘acute’ to complete the sentences. a An

angle is between 90 and 180 degrees.

b An angle of 59 degrees is c An

angle is smaller than a right angle.

d An angle of 100 degrees is e An

100

 .

 .

angle is between 0 and 90 degrees.

8.3 Estimating angles

8.3 Estimating angles Worked example 3

estimate

You can use this decision tree and diagram to help you estimate the size of angles. Is the angle greater than 90 degrees? YES

NO

Is the angle closer to 180 degrees than 90 degrees?

Is the angle closer to 90 degrees than 0 degrees?

YES

NO

YES

NO

The angle must be between 135 and 180 degrees. Look at the angle diagram to make a closer estimate.

The angle must be between 90 and 135 degrees. Look at the angle diagram to make a closer estimate.

The angle must be between 45 and 90 degrees. Look at the angle diagram to make a closer estimate.

The angle must be between 0 and 45 degrees. Look at the angle diagram to make a closer estimate.

A right angle 90 degrees Half a right angle 45 degrees

0 degrees

135 degrees

Two right angles 180 degrees

101

8 Angles

Continued Estimate the size of this angle in degrees.

This angle is greater than 90 degrees.

Use the decision tree first.

It is closer to 90 degrees than 180 degrees. So, it is between 90 degrees and 135 degrees. Looking at the diagram we can estimate that the angle is about 125 degrees.

Then use the diagram to estimate the size of the angle.

Answer: The exact measurement of the angle is 122 degrees. A good estimate would be between 110 degrees and 130 degrees.

Exercise 8.3 Focus 1 How many degrees are in one right angle quarter turn? 2 Circle the best estimate for each angle.

102

a



45 degrees / 100 degrees

b



130 degrees / 80 degrees

102

8.3 Estimating angles

c



95 degrees / 85 degrees

d



30 degrees / 80 degrees

e



110 degrees / 160 degrees

3 Match the label to the angle by estimating the size of the angles. a

b

d

e

c

15 degrees

90 degrees

135 degrees

170 degrees

45 degrees

Practice 4 a How many right angles do you turn to make a full circle? b How many degrees are there in a full circle? c How many right angles do you turn to make three-quarters of a circle? d How many degrees are there in three-quarters of a circle?

103

8 Angles

5 Estimate the size of each angle in degrees. a b

c d

e

f

6 Are you better at estimating acute or obtuse angles? Or are you equally good at both? What can you do to help you make better estimates?

104

8.3 Estimating angles

Challenge 7 a  Sal turned on the spot 270 degrees. How many right angles is that?

b How many more degrees does he need to turn to be back where he started?

8 Describe the relationship between the number of right angles and the number of degrees in an angle.

9 Use a ruler and pencil to draw angles that you estimate are close to these labels. a 90 degrees

b 45 degrees

c 20 degrees

d 135 degrees

e 100 degrees

f 160 degrees

105

9 Comparing, rounding and dividing 9.1 Rounding, ordering and comparing whole numbers Worked example 1 Round 6543 to the nearest 1000. 6543 lies between 6000 and 7000, but it is closer to 7000 than to 6000. 6543 round up 7000 6000

6500

7000

To round to the nearest thousand, look at the hundreds digit: • if it is less than 5 round down • if it is 5 or more round up. 1000s 100s 6

5

10s

1s

4

3

round up to the next thousand Answer: 7000

compare  order  round  round to the nearest

106

9.1 Rounding, ordering and comparing whole numbers

Exercise 9.1 Focus 1 2505 = 2510 rounded to the nearest 10. 2505 2500

2505

2510

Round these numbers to the nearest 10. a 3509 =

to the nearest 10.

b 3489 =

to the nearest 10.

c 4655 =

to the nearest 10.

2 Circle all the numbers that round to 90 when rounded to the nearest 10.

85  94  97  82  86

3 Place these numbers in order of size, starting with the smallest. a 6505  6550  5650  6555  5656

b 1234  2134  2413  1432  2341

4 Use or = to make these number sentences correct. a 8216 

 8126    b 6031 

 6013

5 What are the even numbers that can go in the box?

6160

455110

686400 98 150 >

9.2 Division of 2-digit numbers Worked example 2 a The school cook needs 75 cartons of juice. There are 4 cartons in each pack. How many packs must the cook buy?

Oran g Juice e Orange Juice Orange Juice Orange Juice

division divisor remainder round down

b Amy has 75 cents. One sweet costs 4 cents. How many sweets can she buy?

round up

The first step in both questions is to work out 75 ÷ 4. remainder 3

8 lots of 4

3

0

10 lots of 4 35

75

75 ÷ 4 = 18 r 3 Answer: a You must round up: 18 packs does not give enough cartons.

The cook must buy 19 packs.

b You must round down: Amy does not have enough money to buy 19 sweets.

110

Amy can buy 18 sweets.

9.2 Division of 2-digit numbers

Exercise 9.2 Focus 1 Complete the remainder chart for 24. The first one has been done for you.



24 divided by

2

Remainder

0

3

4

5

6

7

8

9

10

Describe your results.

2 A jug holds 2 litres. Ali needs 26 litres of apple juice for her party. How many jugs of apple juice must Ali make? 2 litres

1 litre

3 Work out these calculations. Show your method. Remember to estimate before you calculate. a 91 ÷ 7



b 96 ÷ 8



111

9 Comparing, rounding and dividing

4 Ali buys 48 tennis balls. The balls are in packs of 3. How many packs does Ali buy?

5 Write the missing numbers.

8÷2=

÷ 4 = 32 ÷

Practice 6 There are 22 pencils in a pot. How many children can have 3 pencils each?

7 Roz bakes 62 cakes. She packs the cakes in boxes. Each box holds 8 cakes. How many boxes does she need to pack all the cakes?

8 Paul wants to put 52 photos in an album. A full page holds 6 photos. He fills as many whole pages as possible. How many photos are left over?

112

9.2 Division of 2-digit numbers

9 Here are four numbers.

65  75  85  95 Divide each number by 7. Circle the number that leaves a remainder of 1.

10 Which pairs of numbers can be written in the boxes?

36 ÷ 

 = 

Challenge 11 Pierre wants to buy a bike. The bike costs $97. Pierre saves $10 every Saturday. How many Saturdays will it take him to save enough money to buy the bike?

12 Use the code to find the name of the planet.



Remainder

1

2

3

4

5

6

7

8

9

Letter

Y

E

T

M

A

R

S

C

U

29 ÷ 5   47 ÷ 3   76 ÷ 7   98 ÷ 9   19 ÷ 10   83 ÷ 7   47 ÷ 2

113

9 Comparing, rounding and dividing

13 Find the rule, then complete these division wheels.

7

9 21

15

36

51

18 45

35

15 85

30

12

14 Draw a line to join each division question to the correct rounding operation. Coconuts cost $2 each. How many coconuts can be bought for $15?

Round up

14 peaches are put in bags. Each bag holds 4 peaches. How many full bags are there? Round down A minibus holds 12 people. 50 people go on an outing. How many minibuses are needed? 15 Which pairs of numbers could be written in the boxes?

114

60 ÷

=

7

10 Collecting and recording data 10.1 How to collect and record data Worked example 1 This data shows the number of bicycles owned by some families. 5, 1, 3, 2, 3, 0, 2, 4, 1, 2, 2, 0, 0 Record the data using a dot plot. Draw an axis long enough to show all the numbers in the set of data. Label the axis. 0

1

2

3

4

It is best to use squared paper to make it easier to space the dots evenly.

5

Number of bicycles

Draw one dot above the number on the axis for each time the number appears in the data.

Answer: 

0

1

2

3

4

5

Number of bicycles

For example, there are two families with 1 bicycle, so show two dots above 1.

data  dot plot  statistical question

115

10 Collecting and recording data

Exercise 10.1 Focus 1 Ingrid counted the birds in her garden each day. Complete the table to show how many days she saw each number of birds.

3, 3, 4, 4, 4, 5, 5, 5, 5, 6, 6, 7, 7, 8 Number of birds

How many days?

3

2

Tip In the list there are two days when Ingrid saw 3 birds. The number 2 goes in the table next to 3 birds.

4 5 6 7 8 2 Billy looked in 15 packets of sweets and counted the red sweets he found in each packet. He used a dot plot to record how many red sweets he found.

a How many packets had 0 red sweets?

b How many packets had 3 red sweets?

0

1

2

3

4

5

Number of red sweets in the packet

116

6

c What is the greatest number of red sweets that Billy found?

10.1 How to collect and record data

3 Billy counted the green sweets in the packets. This is how many green sweets he found in each packet.

1, 1, 2, 2, 2, 2, 3, 3, 3, 4, 6, 6, 6, 7, 8



Complete the dot plot with the number of green sweets he found in the packets.

0

1

2

3

4

5

6

7

8

Number of green sweets in the packets 4 Complete these sentences and choose the words to plan an investigation to find out how many seeds there are in a packet.

Question: How many



I will count / collect how many seeds are in each packet.



I will plant / record the number of seeds in a

are there in a packet?

.

117

10 Collecting and recording data

Practice 5 Aron counted the number of cars that passed his window each hour. a Complete the table to show his results.

10, 11, 14, 10, 10, 13, 14, 13, 13, 13, 11, 13 Number of cars

How many hours?

b Complete the dot plot to show the number of cars that passed each hour.

10

11

Number of

118

___

___

___

10.1 How to collect and record data

6 Use the table or the dot plot in Question 5 to answer these questions. a For how many hours did Aron see 12 cars?

b What was the greatest number of cars Aron saw in an hour?

c For how many hours did Aron see less than 12 cars?

d Did you use the table or the dot plot?



Why?

7 You are going to plan an investigation to find out how many cubes your friends can hold in one hand.



Complete the plan.



Question: How many



People I will use:

 ?

119

10 Collecting and recording data



I will record the data in this table. Number of



Number of

Carry out the investigation and complete the table.

Challenge 8 Nasreen counted the number of seeds in 12 packets of sunflower seeds. This is the number of seeds in each packet:

23, 24, 22, 22, 22, 23, 26, 24, 24, 21, 23, 23 a Draw a table to show how many packets had each number of seeds.

120

10.1 How to collect and record data

b Draw a dot plot to show how many seeds are in each packet.

c How many packets had 25 seeds?

d What was the least number of seeds found in a packet?

e How many packets had more than 23 seeds in them?

f

Reflect on how you answered questions c, d and e. Did you use the table or the dot plot?



Why?

121

10 Collecting and recording data

9 You are going to plan an investigation to find out how many cubes your friends can connect together in a line in one minute.

122



Complete the plan.



Question:



People I will use:



Equipment I will need:



Use the space below to collect your data. Then use the grid below to represent your data in a dot plot.

 ?

11 Fractions and percentages 11.1 Equivalence, comparing and ordering fractions Worked example 1 Put the correct sign < or > between this pair of fractions. _ ​​  3 ​​  

4

equivalent fraction proper fraction

  _ ​​  5 ​​  8

6 ​​  _ ​​  3 ​​   =  ​​ _

Change _ ​​  3 ​​  into an equivalent fraction in eighths. 4

5 ​​  _ ​​  6 ​​   >  ​​ _

Compare the two fractions and write the correct sign.

4 8

8 8

5 ​​  ​​  3 ​​  > ​​ _ Answer: _ 4 8

123

11 Fractions and percentages

Exercise 11.1 Focus 1 Here is a table. The rows for the 1× table and the 5× table are in grey. x

1

2

3

4

5

6

7

8

9

10

1

1

2

3

4

5

6

7

8

9

10

2

2

4

6

8

10

12

14

16

18

20

3

3

6

9

12

15

18

21

24

27

30

4

4

8

12

16

20

24

28

32

36

40

5

5

10

15

20

25

30

35

40

45

50

6

6

12

18

24

30

36

42

48

54

60

7

7

14

21

28

35

42

49

56

63

70

8

8

16

24

32

40

48

56

64

72

80

9

9

18

27

36

45

54

63

72

81

90

10

10

20

30

40

50

60

70

80

90 100



You can use the table to help work out equivalent fractions:



6  ​​ = ​​ _ 9  ​​ = ​​ _ 10 ​​  3  ​​ = ​​ _ 5  ​​ = ​​ _ 8  ​​ = ​​ _ 2  ​​ = ​​ _ 4  ​​ = ​​ _ 7  ​​ = ​​ _ _ ​​  1 ​​  = ​​ _



Use the table to complete these equivalent fractions.

5

10

15

20

25

30

35

40

45

50

2   ​  a _ ​  1 ​  = ​ _ = ​ _  ​   = _ ​  4   ​  = ​ _  ​   = _ ​  6   ​  = ​ _  ​   = _ ​  8   ​  = ​ _  ​   = _ ​  10  ​  14 2 6 10 18

124

11.1 Equivalence, comparing and ordering fractions

9   ​  15  ​  21  ​  27  ​  b _ ​  3 ​  = ​ _  ​   =​ _ = ​ _  ​   =​ _ = ​ _  ​   =​ _ = ​ _  ​   =​ _ = ​ _  ​    4 8 16 24 32 40

16  ​  8   ​  4   ​  12  ​  c _ ​  2 ​  =​ _ = ​ _  ​   =​ _ = ​ _  ​   =​ _ = ​ _  ​   =​ _ = ​ _  ​   = _ ​  20  ​  15 3 9 21 27



Use the fraction wall to help you answer questions 2 and 3. 1 1 2

1 2 1 3

1 3

1 4

1 4

1 4

1 6 1 8

1 3

1 6 1 8

1 6 1 8

1 8

1 4

1 6 1 8

1 6 1 8

1 6 1 8

1 8

2 Write each set of fractions in order, smallest first. Write down the equivalent fractions you use to work out each answer. 5 ​​   1 ​​ , ​​ _ _ a ​ ​  2 ​ ​, _ ​​  3 ​​ , ​​ _ 3 4 3 8 1 ​​ , ​​ _ 7 ​​   _ b ​ ​  2 ​ ​, _ ​​  5 ​​ , ​​ _ 3 6 2 8 3 Put the correct sign < or > between each pair of fractions. _ a ​ ​  1 ​ ​  2

5 ​​   ​​ _ 8

_ b ​ ​  2 ​ ​  3

5 ​​   ​​ _ 6

125

11 Fractions and percentages

Practice

4 Colour in red the squares containing fractions equivalent to _ ​​ 2 ​​ . 3 What letter can you see?

Colour in blue the squares containing fractions equivalent to _ ​​ 3 ​​ . 4 What letter can you see? _ ​​  8  ​​ 

__ ​​  5 ​​ 

_ ​​  18 ​​ 

_ ​​  6  ​​ 

_ ​​  9  ​​ 

_ ​​  15 ​​ 

_ ​​  30 ​​ 

_ ​​  2 ​​ 

_ ​​  10 ​​ 

_ ​​  10 ​​ 

_ ​​  10 ​​ 

_ ​​  18 ​​ 

_ ​​  3  ​​ 

_ ​​  21 ​​ 

_ ​​  14 ​​ 

_ ​​  6 ​​ 

_ ​​  22 ​​ 

_ ​​  30  ​​ 

_ ​​  12 ​​ 

_ ​​  27 ​​ 

_ ​​  3 ​​ 

_ ​​  20 ​​ 

_ ​​  1 ​​ 

_ ​​  16 ​​ 

_ ​​  9  ​​ 

_ ​​  24 ​​ 

_ ​​  1 ​​ 

_ ​​  6  ​​ 

_ ​​  4 ​​ 

_ ​​  20 ​​ 

_ ​​  12 ​​ 

_ ​​  2 ​​ 

_ ​​  6 ​​ 

_ ​​  4  ​​ 

_ ​​  18 ​​ 

12 3

21 30 6

6

12 9

2

24

27 15

33

24 18

20 25

100 36 5

12

20

24

10

16

36 4

32 8

10

40

28 4

12 36

5 Here are three fractions.

1 ​​   ​​  2 ​​  _ _ _ ​​  2 ​​   ​​     

4 5 3 Write each fraction in the correct box on the number line. 1 2

0

1

6 Which is larger? a 5 or 3  8 4

126

b 5 or 2  6 3

c 1 or 3​​  2 8

11.1 Equivalence, comparing and ordering fractions

7 Choose different numbers for each numerator to make this number sentence correct.

8

>

4

How many different answers can you find? Both fractions must be less than 1.

Challenge 8 Find sets of three equivalent fractions. _ ​​  3 ​​ 

_ ​​  5 ​​ 

_ ​​  6  ​​ 

_ ​​  8  ​​ 

_ ​​  4  ​​ 

_ ​​  2 ​​ 

_ ​​  10 ​​ 

_ ​​  6 ​​ 

_ ​​  10 ​​ 

_ ​​  3  ​​ 

_ ​​  20 ​​ 

_ ​​  4 ​​ 

_ ​​  30  ​​ 

_ ​​  9  ​​ 

_ ​​  2 ​​ 

4

3

24



6

12 6

20 8

100

12 25

12

10

10 5

List your sets here:

127

11 Fractions and percentages

9 Which is the odd one out? Explain your answer. 10 ​​   ​​  2 ​​  _ _ _ ​​  9  ​​   ​​     



15

12

3

Find more than one answer.

10

and



is a multiple of 5



is a multiple of 6

stand for two different numbers.

=2 3



What numbers could

and

stand for?

3 ​​ . 11 Circle the fraction that is greater than _ ​​ 1 ​​  but less than ​​ _ 4 2 5 ​​   ​​  2 ​​   ​​  7 ​​  _ _ _ _ ​​  3 ​​   ​​       



8

4 ​​  is greater than ​​ _ 4 ​​ .’ 12 Alana says, ‘​​ _ 7 9 Is Alana correct?

128

Explain how you know.

4

8

8

11.2 Percentages

11.2 Percentages Worked example 2 a Shade 10% of each grid in grey. b

Shade _ ​​  1 ​​  of each grid in black.

percent percentage

4

a 10% = 10 out of 100

There are 100 squares in each grid. Shade 10 squares grey.

1 ​​  = 25% = 25 out of 100 b ​​ _ 4 Answer:

Shade 25 squares black. It does not matter which squares you shade as long as you shade the correct number.

129

11 Fractions and percentages

Exercise 11.2 Focus 1 Write the percentage that is coloured in grey.

2 Colour these diagrams to show the following percentages. a 50%



130

11.2 Percentages

b 75%

c 10%

3 Write these fractions as percentages. a 35 out of 100 =

 %

b 36 out of 100 =

 %

c 72 out of 100 =

 %

d 14 out of 100 =

 %

e 67 out of 100 =

 %

131

11 Fractions and percentages

Practice 4 Colour these diagrams to show the following percentages. a 55%

b 48%



132

11.2 Percentages

c 1%

5 Join pairs of shapes that make a whole circle. You will need to work out the percentage for one of the circles.

% 67%

75%

25%

20%

33%

50% 50%

133

11 Fractions and percentages

6 Zhen looks at the label in her coat. The coat is made of wool and cotton. Part of the label is missing. 80% wool % cotton



What percentage of the coat is cotton?

Challenge 7 Write the percentage of each shape that is coloured in.

%

%

%

%

8 Eighty percent of the learners in Year 4 won a prize for good attitude in all lessons. There are 100 learners in Year 4. How many learners won a prize?

134

11.2 Percentages

9 The table shows the colours of flowers in a garden. Colour

Percentage of flowers

red yellow

50%

pink white

20%

The percentage of red and pink flowers is the same. What percentage of the flowers are pink?

10 Colour 25% of each of these grids. a

b

135

12 Investigating 3D shapes and nets 12.1 The properties of 3D shapes Worked example 1 Draw arrows from the words to the parts of the shape. Face Edge Vertex Answer: Face Edge Vertex

Faces are the flat surfaces of a 3D shape. This rectangle is a flat surface on the hexagonal prism. Edges are the lines where two faces meet. This line is where two faces meet on the hexagonal prism. Vertices are the points where edges meet. Three edges meet at this point on the hexagonal prism.

cone edge face prism pyramid tetrahedron vertex / vertices

136

12.1 The properties of 3D shapes

Exercise 12.1 Focus 1 Draw arrows from the words to the parts of this shape.

Face Edge Vertex

2 This is a cuboid.



Mark the edges with a green line.



Mark the vertices with a red circle. a How many edges are there? b How many vertices are there? c How many faces are there?

137

12 Investigating 3D shapes and nets

3 a This is a pentagon-based pyramid.



Name the shaded face.

b This is a triangular prism.



Name the shaded face.

4 Write the names of the seven faces of a pentagonal prism

138



   

   



   

   

   

12.1 The properties of 3D shapes

Practice 5 Complete these sentences about the tetrahedron.

A tetrahedron has:



faces,

edges and

vertices.

6 This is a face from one of these shapes.



Tick the shapes it could belong to. A

B

D

C

E

7 Name a 3D shape with more than two triangular faces.

8 Name a 3D shape with a circular face.

139

12 Investigating 3D shapes and nets

Challenge 9 Complete the table to show how many faces, edges and vertices each shape has. Shape

Number of faces

Cuboid

6

Number of edges

Number of vertices

12

Triangular prism

6

Pentagon-based pyramid Hexagonal prism 5

8

5

10 This is a face from one of the shapes in the list.



Tick the shapes it could belong to.



Cube 



Hexagonal prism 

      Triangular prism 

      Tetrahedron 

      Square-based pyramid 

11 Describe how triangular prisms and triangle-based pyramids are similar and different.

140

12.2 Nets of 3D shapes

12.2 Nets of 3D shapes Worked example 2

net tetrahedron

What shape does this net make? A A tetrahedron B A square-based pyramid C A cube

First label the number of faces and the shape of the faces.

triangle

triangle

3

1 4 square

triangle

2

5 triangle This net has 5 faces. It has 1 square face and 4 triangular faces.

The net cannot make a cube.

This net has 1 square face and 4 triangular faces. The square face must be the base.

Check if the shape will be a prism or a pyramid. • A prism has two opposite faces connected by rectangles. • A pyramid has a base shape and all other faces are triangles that meet at a vertex.

Answer: B. This is the net of a square-based pyramid.

141

12 Investigating 3D shapes and nets

Exercise 12.2 Focus 1 Circle the net that will make a hexagonal prism.

A

B

2 All of these nets are the same except one. Circle the odd one out. A

142

B

C

D

12.2 Nets of 3D shapes

3 Circle the shape that can be made with this net.



   Cube        Triangular prism     Pentagon-based pyramid

     

      

Practice 4 Copy this net onto card and cut it out. Fold it into a 3D shape. What 3D shape have you made?

143

12 Investigating 3D shapes and nets

5 Circle the net that will make a triangular prism. A

B

C

6 Draw lines to match the shape to its net. A

cube

B

cylinder C

cone

144

12.2 Nets of 3D shapes

7 Tick the shape that can be made with this net.

A cube 



An octagonal prism 



An octagon-based pyramid 

Challenge 8 Draw lines to match the shape to its net. cuboid

A

B tetrahedron

C

heptagonal prism

D square-based pyramid

145

12 Investigating 3D shapes and nets

9 Arnold has drawn this net to make a pentagon-based pyramid. Explain how you know that this net will not make a pentagon-based pyramid.

10 Write the name of the shape that this net makes.

146

13 Addition and subtraction 13.1 Adding and subtracting efficiently Worked example 1

carry efficient

Work out 367 + 185. Method 1:

• Write the numbers in columns.

100s 10s +

0s

3

6

7

1

8

5

1

2

1

4

0

4

0

0

5

5

2

Method 2: 100s 10s +

0s

3

6

7

1

8

5

5

5

2

1

1

• Add the ones, then the tens, then the hundreds. • 7 + 5 = 12 • 60 + 80 = 140 • 300 + 100 = 400 • Add these totals to get the answer: 400 + 140 + 12 is 552

You can write the calculation more efficiently by ‘carrying’ the tens and hundreds. • 7 + 5 is 12. Record 2 ones and carry 1 ten. • 60 + 80 + 10 is 150. Record 5 tens and carry 1 hundred. • 300 + 100 + 100 is 500. Record 5 hundreds.

Answer: 552

147

13 Addition and subtraction

Exercise 13.1 Focus 1 There are 215 girls and 259 boys in the school hall. How many boys and girls are in the hall altogether?

2 Look at this diagram. Write an estimate.

Here is a subtraction. 89 – 31

90 – 30

58

60

Check that the answer is close to your estimate.

Subtract for your estimate.

Find the answer.

Do the same for these calculations. a 128 − 62

b 358 − 254

c 548 − 392







3 Chloe has three digit cards. 3

148

 

4

 

9

She makes the largest number she can with the three cards. Then she makes the smallest number she can with the three cards. What is the difference between the two numbers Chloe makes?

13.1 Adding and subtracting efficiently

4 649 people go to a concert. 290 people are women. 312 people are men. How many children are at the concert?

Practice

Birmingham

179 188 127 334

Cardiff

179

London

188 269

269 278 489 298 441

Manchester 127 278 298 Newcastle

Newcastle

Manchester

London

Cardiff

Birmingham

5 The table shows the distance in kilometres between five towns in the United Kingdom.

212

334 489 441 212

Tip Make sure you can read the table correctly by finding the distance from London to Cardiff. You should find it is 269 kilometres.

a Nami travels from Birmingham to Manchester and then from Manchester to Newcastle. How many kilometres does she travel altogether?

b Raphael travels from London to Birmingham and then from Birmingham to Newcastle. How many kilometres does he travel altogether?

149

13 Addition and subtraction

6 Subtract the largest 2-digit number from 200.

7 Use only the digits 4 and 5 to complete this calculation. You may use the digits more than once. +

= 500

8 Darius spilt ink on his work. His answer is correct. What is the missing digit? 7

3

2

– 3

8

9

4

1

3

Challenge 9 Write the missing digits. 5 +

1

9 9

3

4

2

10 Use each of the digits 4, 5, 6, 7, 8, 9 in each calculation to make these number sentences true.

150

–

= 333

–

= 111

13.1 Adding and subtracting efficiently

11 What total closest to 800 can you make using two of these numbers? 605 395

   

402 899

   

503 901

   

789 197

12 Last year there were 82 girls and 93 boys in Year 4. This year there are 190 students in Year 4. Of these students, 95 are boys. How many more girls are in Year 4 this year than last year?

   13 Find five pairs of numbers that add up to 900.



One has been done for you.



672 + 228 = 900

545

238

86

228

96

791

355

601

672

109

589

437

463

322

814

465

151

13 Addition and subtraction

13.2 Adding and subtracting fractions with the same denominator Worked example 2 Find the missing fraction. 1  ​  + ​ _ ​​  3  ​  + ​ _ 11

=_ ​​  14 ​​  11

11

1  ​​ = ​​ _ 4  ​​  Calculate _ ​​  3  ​​ + ​​ _ 11 11 11 The number sentence can then be written as 4  ​​ + ​​ _ 11

So

=_ ​​  14 ​​  11 14 ​​ . =_ ​​  10 ​​ which can be found by counting on from _ ​​  4  ​​ to  ​​ _ 11 11 11

Answer: _ ​​  10 ​​  11

Exercise 13.2 Focus

1 Jamila has cycled _ ​​ 3 ​​  of the way to the gym. 4 What fraction of the distance does she still have to go?

152

improper fraction proper fraction

13.2 Adding and subtracting fractions with the same denominator

2 Draw lines between two fractions that total 1. Find all the pairs that total 1. 2 5

2 8

3 4

3 5 4 5

1 4

6 8

5 6 1 5

1 6

3 Work out these calculations. 3 ​​   = a _ ​​  5 ​​  + ​​ _ 6 6 3  ​​  = b _ ​​  7 ​​  − ​​ _ 8 8 3  ​​   = c _ ​​  11 ​​ − ​​ _ 12 12 5  ​​   = d _ ​​  7  ​​ − ​​ _ 12 12 4 Write the missing numbers. a _ ​​  3 ​​  + 8

6 ​​  − =_ ​​  9 ​​      b ​​ _ 8 8

=_ ​​  1 ​​  8

5 Find the missing fraction.

1 ​​  + _ ​​  3 ​​  + ​​ _ 8

8

=_ ​​  7 ​​  8

153

13 Addition and subtraction

Practice 6 Write the missing numbers.

6 ​​  = _ ​​  2 ​​  + ​​ _

3 ​​  + ​​ _ 3 ​​  = ​​ _ 4 4

5 ​​  + ​​ _ 7 ​​  = ​​ _ 9 9



2 ​​  = _ ​​  2 ​​  + ​​ _

6 ​​  = 7 ​​  + ​​ _ ​​ _ 8 8

3 ​​  + ​​ _ 4 ​​  = ​​ _ 6 6



4 ​​  − ​​ _ 1 ​​  = ​​ _

6 ​​  − ​​ _ 3 ​​  = ​​ _ 9 9

5 ​​  − ​​ _ 2 ​​  = ​​ _ 7 7



1  ​​ = _ ​​  9  ​​ − ​​ _

4  ​​ − ​​ _ 1 ​​  = ​​ _ 9 9

7 ​​  − ​​ _ 2 ​​  = ​​ _ 8 8

7

3

5

10

7

3

5

10

7 Write the missing fraction.

+

3 5

=

4 5

+

=

8 Draw lines between two fractions that total _ ​​ 10 ​​.  9 10 _ Find all the pairs that total ​​   ​​.  9 2 9 5 9

8 9

7 9

5 9 3 9

4 9

154

6 9

13.2 Adding and subtracting fractions with the same denominator

9 Find different ways to solve this calculation.

= 4 3

+

10 Write the missing numbers. a _ ​​  7 ​​  − 9

=_ ​​  2 ​​   9

b 

3 ​​  −_ ​​  2 ​​  = ​​ _ 7 7

Challenge 11 Find the missing numbers.

_ ​​  2 ​  +​

7

5 ​​   +_ ​​  2 ​  =  ​ _ 3 3







3 ​  +​ _ =_ ​​  6 ​​ ​​    7 4

_ ​​  8 ​​  −

9

6 ​​  − _ =_ ​​  4 ​​ ​​    9 9 1 ​​   −_ ​​  2 ​​  = ​​ _ 5 5

5 ​  +​ _ =_ ​​  5 ​​ ​​    4 9 7 ​​   +_ ​​  6 ​  =  ​ _ 8 8 11 ​​ − _ =_ ​​  3 ​​ ​​    9 12 1 ​​   −_ ​​  5 ​​  = ​​ _ 7 7

=_ ​​  10 ​​  9 7 ​​  +_ ​​  3 ​  =  ​ _ 6 6 =_ ​​  2  ​​  12 3 ​​  −_ ​​  4 ​​  = ​​ _ 9 9

155

13 Addition and subtraction

12 Find different ways to solve this calculation.

= 7 3

+

7 ​​ . 13 a Write two fractions that have a sum of ​​ _ 8 3 ​​ . b Write two fractions that have a difference of ​​ _ 8

14 Find the missing fraction.

3 ​  + ​ _ 2 ​  + ​ _ 1 ​​  −_ ​​  2 ​  = ​ _ 9 9 9 9

15 In the diagram the fraction in each box is the sum of the two fractions below it. Write the missing fractions. 21 20 9 20 7 20

156

14 Area and perimeter 14.1 Estimating and measuring area and perimeter Worked example 1

area perimeter

Measure and calculate the perimeter of this shape.

Place your ruler carefully along each side of the shape and record the length of each side. Remember the units.

  

  

The sides are 3 cm, 4 cm and 5 cm. 3 + 4 + 5 = 12

Add together the side lengths to find the perimeter.

Answer: The perimeter of the shape is 12 cm.

157

14 Area and perimeter

Exercise 14.1 Focus 1 Add the lengths of the lines to find the perimeter of the rectangle. 1 ________ cm 4 ________ cm

2 ________ cm 3 ________ cm



Line 1 =

cm



Line 2 =

cm



Line 3 =

cm



Line 4 =

cm



Perimeter =

cm

2 Add up the lengths of the castle walls to find the perimeter in metres. 14 m

12 m

Castle

14 m

158

12 m

14.1 Estimating and measuring area and perimeter

3 Draw and shade three more different shapes on this centimetre square paper that have an area of 8 cm2. 1 2 3

6

4

7

5

8

4 Number the squares that have over half shaded to estimate the area. Write the area in cm2.

A

B 1 cm

a Area of shape A = b Area of shape B = c Add the areas together to estimate the total shaded area.

Total area is

159

14 Area and perimeter

Practice 5 Measure and record the perimeter of each rectangle. Remember to use the correct units. a

c

160

b

14.1 Estimating and measuring area and perimeter

6 Add up the lengths around the park to find out how long the perimeter fence is. Remember to use the correct units. 15 m

10 m 20 m

8m 12 m

The perimeter fence is

7 Divide this shape into four sections so that each section has an area of 9 cm2.

161

14 Area and perimeter

8 Number the squares that are over half shaded to estimate the area. Write the area in the correct square units.

A

1 cm

B

a Area of shape A is b Area of shape B is c Add the areas together to estimate the total shaded area.

Total area is

Challenge 9 Measure and record the perimeter of this shape in millimetres.

162

14.1 Estimating and measuring area and perimeter

10 Estimate the total shaded area.

1 cm



The total shaded area is approximately

11 On this centimetre square paper draw and shade four different shapes with curved sides that each have an estimated area of 6 cm2.

163

14 Area and perimeter

12 What is the perimeter of the castle wall? 20 m 6m

5m

10 m

14 m

Castle

6m

5m 16 m

164

14.2 Area and perimeter of rectangles

14.2 Area and perimeter of rectangles Worked example 2 Find the area of this rectangle. 5 cm

5 cm

5 10

Count the squares to find the area. This rectangle has 5 squares in each row so you can count the squares in 5s.

15 20 25 Answer: The area is 25 cm2.

165

14 Area and perimeter

Exercise 14.2 Focus 1 Join the dots carefully with a ruler.

a What shape have you made? b Measure and label each side in centimetres to show how long it is. c What is the perimeter of your shape? 2 a How many squares in each row of this rectangle?

b Count in 3s to find out how many squares altogether. 3 6 9



squares

c Measure one side of a square in the rectangle.

1 square is

wide.

d Circle the correct area for the rectangle.

166

18 mm2   18 cm2   18 m2   18 km2

14.2 Area and perimeter of rectangles

3 Look at this rectangle.

a How many squares in each row? b Use a ruler to measure the length of the rectangle.

The rectangle is

cm long.

c How many squares are there in each column? d Use a ruler to measure the width of the rectangle. wide.

The rectangle is

e Use these words to complete the sentence. length



       

row

       

squares

The

of the rectangle is the same as the number of

in a

.

167

14 Area and perimeter

Practice 4 In this space draw a rectangle that is 8 cm long and 5 cm wide. The distance between the dots is 1cm.



What is the perimeter of your rectangle?

5 a How many squares are there in each row of this rectangle?



squares

b Circle the correct area for the whole rectangle.

168

8 cm2   36 mm2    28 cm2    32 m2    32 cm2    24 km2

14.2 Area and perimeter of rectangles

6 Complete the sentences to describe the rectangle.



There are



The rectangle is



squares in each row and cm long and

rows of

makes

squares in each column. wide. squares altogether.



The area of the rectangle is

multiplied by



The area of the rectangle is

cm2.

 .

7 Calculate the area and perimeter of these rectangles using the measurements shown. Remember to record the units. a

8m

3m



Area is

b

    Perimeter is 5 km

2 km



Area is

    Perimeter is

169

14 Area and perimeter

c

10 mm

8 mm



Area is

d

    Perimeter is 7 cm

7 cm



Area is

    Perimeter is

Challenge 8 Draw a rectangle that is 5 cm long and 4 cm wide. Use a ruler.

170



Find the perimeter and area.



Perimeter =

   Area =

14.2 Area and perimeter of rectangles

9 Without measuring, work out the missing lengths from these rectangles. Write in the missing lengths and work out the perimeter of each rectangle. a

b

3m 1m

9 km

Perimeter is 6 km

c



72 mm

Perimeter is

6 mm

Perimeter is

10 Explain in words how you can use the measurements for the length and width of a rectangle to calculate its area.

171

14 Area and perimeter

11 Calculate the area of these rectangles using the measurements shown. a



b

5 km 9 mm

12 km

Area is





c

11 mm Area is d 2m

8 cm

1

32 m

8 cm



172

Area is



Area is

15

Special numbers

15.1 Ordering and comparing numbers negative number  order

Worked example 1

Write these temperatures in order starting with the lowest temperature. −14 °C   4 °C   14 °C   −1 °C   0 °C   −8 °C Think about where the numbers go on a number line. –15

–10

–5

0

5

10

15

Order the negative numbers first, then zero, then the positive numbers. Negative numbers are always smaller than positive numbers.

Answer: −14 °C, −8 °C, −1 °C, 0 °C, 4 °C, 14 °C

173

15 Special numbers

Exercise 15.1 Focus 1 Write each set of temperatures in order starting with the coldest temperature. Use the number line to help you. Remember that as you move to the right (→) numbers get larger. –10 –9 –8 –7 –6 –5 –4 –3 –2 –1 0

1

2

3

4

5

6

7

8

9 10

a 0 °C   −9 °C   3 °C   −2 °C

b 3 °C   −4 °C   7 °C   −8 °C

c −2 °C   2 °C   7 °C   −10 °C

2 a Write these numbers in order starting with the smallest.

−5   5   10   −15   0   −10

b Describe the number pattern you have made.

c If you continue the pattern, will you write 71? How do you know without writing all the numbers?

Tip 3 Which is larger −4 or −1? Write your answer using one of the symbols > or


>

10 The table shows the average temperatures in some cities in January. City

Temperature (°C)

Bejing, China

 −3

Budapest, Hungary

  1

Delhi, India

 14

Istanbul, Turkey

  5

Karachi, Pakistan

 18

Moscow, Russia

 −8

Ulaanbaatar, Mongolia

−20

a Which is the coldest place?

b Which is the warmest place?

c Put the temperatures in order starting with the coldest.

11 Compare each pair of numbers using the symbols > or < .

176

−3

−4   −19

11   0

−1

15.2 Working with special numbers

12 A sequence starts at 50, and 8 is subtracted each time.

50   42   34 . . .



If the sequence continues in the same way, what are the first two numbers less than zero?

15.2 Working with special numbers even factor multiple  odd square number

Worked example 2 Here are four labels. multiple of 5

  

not a multiple of 5

  

odd

  

even

Write the labels on the sorting diagram.

30

5 50

35

18

7 6

11

even

Work systematically.

odd

30

5 50

18

35 7

6

11

Look at all the numbers in the first column and place the label. All the numbers in the first column are even. Then look at all the numbers in the second column. All the numbers in the second column are odd. Repeat for each row.

177

15 Special numbers

Continued Answer: even

odd

30

multiple of 5

5 50

35

18

not a multiple of 5

7 6

11

Exercise 15.2 Focus 1 Colour all the odd numbers. What is the hidden number? 416 636 50

32 412 806 154

232 135 220 53 861 657 72 198 687

8

100 654 423 98

110 909 68 851 595 677 86 404 45 676 53 358 730 590 206 701 294 825 117

5

358

682 566 742 174 552 340 246

You could use squared paper to make similar puzzles for your friends to try.

2 Look at the number grid.

178

Circle all the multiples of 7. 61

62

63

64

65

66

67

68

69

70

71

72

73

74

75

76

77

78

15.2 Working with special numbers

3

stands for a missing number.



Write the missing number and missing words in these number sentences.



3 and 7 are



is a

of

because 3 × 7 =

of 3 and 7 because

÷ 3 = 7 and

÷7=3

4 Put a cross (X) through the number that is in the wrong place. Write it in the correct place.

multiples of 5 50

45

55

numbers greater than 50 62

42 5 Write all the factor pairs for 24.

×

= 24   

×

= 24



×

= 24   

×

= 24

6 Kojo says, ‘Nine is a square number.’ Is Kojo correct? Explain how you know.

179

15 Special numbers

Practice 7 Kofi is thinking of a number. • It is a multiple of 3. • It is a multiple of 5. • It is an odd number. • It is between 20 and 60.

What is Kofi’s number?

8 Explain why a number ending in 5 cannot be a multiple of 4.

9 Here is a diagram for sorting numbers. Write the numbers 24, 25, 64 and 65 in the correct box on the diagram.

n eve are

squ

start

ber

num

not

a numsquar ber e

odd

n eve odd

10 a Write all the factors of 20.

b Write all the multiples of 20 that are less than 100.

180

15.2 Working with special numbers

11 Use four different square numbers to make these calculations correct. +

=5

+

= 25

Challenge 12 Write the 2-digit odd number that is a multiple of 7. 4

What is special about this number?

13 Here are four digit cards.

3

  

4

  

5

  

6

Use each of these cards to make a total that is a multiple of 6. Each card can only be used once. and

14 Look at these numbers and complete the sentences.

8

  

12

  

25

  

40

a 8 is the odd one out because

b 12 is the odd one out because

181

15 Special numbers

c 25 is the odd one out because

d 40 is the odd one out because

15 Use this tree diagram to sort a set of whole numbers. You could use 18, −24, 27, 19, 14, −21, −5, −14, or choose numbers of your own. Is it a negative number? Yes Is the number even?

Yes

Is the number less than –20?

Yes

182

No

No

Is the number even?

Yes

Is the number less than –20?

Yes

No

No

No

Is the number a multiple of 9?

Yes

No

Is the number a multiple of 9?

Yes

No

15.3 Tests of divisibility

15.3 Tests of divisibility Worked example 3

divisibility rule  divisible

Use all the digits 8, 0, 9 and 5 to make the smallest 4-digit number that is divisible by both 2 and 5. A number that is divisible by both 2 and 5 must be divisible by 10. The number will have 0 in the ones place. Arrange the other digits to make the smallest possible number. Answer: 5890

Exercise 15.3 Focus 1 Circle all the numbers that are divisible by 2.

232  234  243  223  251  215

2 Write these numbers in the Venn diagram. 302  25  203  400  205  52  502

divisible by 2

divisible by 5

183

15 Special numbers

3 Draw a line to complete the divisibility rules.

Divisible by 2



ones digit is 0 or 5



Divisible by 5



tens and ones digits are 0



Divisible by 10



ones digit is 0, 2, 4, 6 or 8



Divisible by 100

ones digit is 0

Practice 4 Use one of these numbers to complete each sentence.

2  5  10  100 a A number is divisible by

if the ones digit is 0, 2, 4, 6 or 8.

b A number is divisible by

if the ones digit is 0.

5 Write these numbers in the table.

25  500  310  1000  105  690 Divisible by 5

Divisible by 5 and 10

Divisible by 5, 10 and 100

6 Colour every number that is divisible by 2, 5 or 10. 1 70 20 80 3 13 61 17 43 52 54 90 31 27 4 63 32 69 39 44 19 29 75 9 14 59 67 62 46 10 53 22 70 25 7 12 28 55 73 63 8 17 34 29 77 32 71 43 59 49 62 79 41 30 38 34 73 33 51 51 69 53 57 105 87

184

What is the answer to the secret calculation that appears when you have coloured the numbers?

15.3 Tests of divisibility

Challenge 7 Alex says, ‘The number fifteen thousand five hundred and fifty-three is divisible by 5.’ Is he correct? Explain how you know.

8 Which of these numbers is divisible by 5 but not by 2 or 10?

250  205  502  520



How do you know?



Write down two more numbers that are divisible by 5 but not by 2 or 10.

9 Here is a set of numbers.

20, 50, 75, 300, 350, 600, 650, 675 a Write all the numbers that are divisible by 100.

b Write all the numbers that are divisible by 50.

c Write all the numbers that are divisible by 25.

10 What is the smallest number that can be added to 333 to make it divisible by 25?

185

16 Data display and interpretation 16.1 Displaying and interpreting data Worked example 1 Display the data in this frequency table using a bar chart. Frequency table showing how many pets each person has Number of pets

Tip

Number of people

0

10

1

13

2

 7

3

 4

4

 2

Always use a ruler and work neatly so that the bar chart will be easy to read and interpret.

Answer:

186

14 12 10 8 6 4 2 0

Number of people

Choose a scale for the vertical axis. The highest number in the data is 13, so the axis must go up to at least 13. The data will be clearest if it is labelled in 1s or 2s.

Bar chart showing how many pets each person has

0

1 2 3 Number of pets

4

Add a title. The title explains what the data is about.

Draw bars, using the scale to match the data in the table to the data in the bar chart.

Label the horizontal axis and the vertical axis. The headings in the table can be used to label the axes.

16.1 Displaying and interpreting data

bar chart  Carroll diagram  pictogram  Venn diagram

Exercise 16.1 Focus 1 Label each diagram, chart or graph with its name.

a

pictogram

Venn diagram

Carroll diagram

bar chart

dot plot

frequency table

Sweet colour

Number in the packet

b

red yellow green key:

0

2

3

4

5

Number of pets

= 2 sweets

c

1

Even

Not even

d

Coin

Number of coins

Multiple of 10

5 cents

4

Not a multiple of 10

10 cents

1

25 cents

2

50 cents

1

187

16 Data display and interpretation

2 Circle the chart or diagram you should use for displaying each set of data. Explain why you would use that chart or diagram. a Sorting the colour and name of a set of shapes. I would use a

Pictogram / Carroll diagram because

 .

b Showing how many people voted for different songs. I would use a

Venn diagram / Bar chart because

 .

3 Sort the numbers from 1 to 20 into this Venn diagram.

Less than 10

Odd

4 Choose your own categories to sort shapes in this Carroll diagram. Choose a colour _____________

Choose a shape name

not _____________

______________ not ______________

188

Draw and colour two shapes in each section to match the properties you have chosen.

16.1 Displaying and interpreting data

5 Display the data in this frequency table using a bar chart. Frequency table showing how many pets each person has Number of pets



Number of people

0

20

1

14

2

9

3

3

4

1

You can use Worked example 1 to help you.

a How many people have two pets? b How many people have more than two pets?

189

16 Data display and interpretation

Practice 6 Name the type of diagram, chart or graph. a

b

Number of vehicles

Multiples of 2

10

Multiples of 5

Multiples of 10

8 6 4 2 0 Buses Cars Vans Type of vehicle

7 a Name two charts or diagrams you could use for sorting a set of numbers by their properties.

b Name two charts or diagrams you could use for showing how many stamps have been collected by each person in a group.

8 Sort the numbers from 1 to 30 into this Venn diagram. Odd

Less than 20

190

Multiples of 5

16.1 Displaying and interpreting data

9 Four girls are sorted into this Carroll diagram.



Curly hair

Not curly hair

Wearing glasses

Amira

Bibi

Not wearing glasses

Clara

Delia

Draw a picture of Clara.

10 You are going to investigate the length of the names of people you know.

Write 30 first names of people that you know. You can write one name in each box.



Mark the length of each name on this tally chart and find the frequency for each group of lengths. Number of letters in the name

Tally

Frequency

1 to 3 4 to 6 7 to 9 10 or more

191

16 Data display and interpretation



Choose a type of graph or chart to represent your data.



What type of graph or chart will you use?



Why will you use that type of graph or chart?



192

 .



Use this space to draw your graph or chart.



Write two sentences to describe what you have found out about the length of first names of people you know.



1

 .



2

 .

16.1 Displaying and interpreting data

11 Daisy counted the birds she saw outside her window for one hour each day for five days. The bar chart shows her results. Number of birds Daisy saw each day 20 18 Number of birds

16 14 12 10 8 6 4 2 0

1

2

3 Day

4

5



Kwame counted the birds he saw outside his window for one hour each day for five days. This frequency table shows his results.



Number of birds Kwame saw on each day Day



Number of birds

1

10

2

 7

3

 7

4

 4

5

 3

Draw a bar chart of the birds Kwame saw. Use the same scale as the bar chart for the birds that Daisy saw.

193

16 Data display and interpretation

a Describe one similarity between the two sets of data.

b Describe one difference between the two sets of data.

c Explain a possible reason for the differences in the data.

Challenge 12 a Explain what a Carroll diagram might be used for.

b Explain what a pictogram might be used for.

194

16.1 Displaying and interpreting data

13 a Circle the numbers that are in the wrong place in this Venn diagram. 11 16 18 19

Odd 17

15

13

20 5

7 9

Less than 10 8 10

1 3

6

2

14 4

Factor of 12 12

b Explain why one of the sections has no numbers in it.

14 Choose your own categories to sort 3D shapes in this Carroll diagram.

Write the names of at least one 3D shape in each section to match the properties you have chosen. If you cannot find a shape to match a section you will need to change your headings.

195

16 Data display and interpretation

15 Anna timed how long it took her to get to school each day for five days. This frequency table shows how long it took Anna to get to school each day. Day



1

25

2

17

3

19

4

23

5

28

Carlos timed how long it took him to get to school for the same five days. This frequency table shows how long it took Carlos to get to school. Day

196

Number of minutes to get to school

Number of minutes to get to school

1

11

2

 9

3

11

4

15

5

18

16.1 Displaying and interpreting data



Draw two bar charts on the grid provided to display the time it took Anna and Carlos to get to school.

a Why is it useful to use the same scale for both bar charts?

b Describe one similarity between the two sets of data.

c Describe one difference between the two sets of data.

d Explain a possible reason for the differences in the data.

197

17 Multiplication and division 17.1 Developing written methods of multiplication Worked example 1

estimate product

Ingrid works in a garden centre. She plants 4 seeds in each cell in a tray. Each tray has 270 cells. She fills every cell. How many seeds does Ingrid plant altogether? Ingrid works in a garden centre.

Read the problem carefully.

She plants 4 seeds in each cell in a tray.

Underline the important information.

Each tray has 270 cells. She fills every cell. You need to work out 270 × 4

Decide what calculation to do.

Estimate:

Estimate the answer.

200 × 4 = 800 300 × 4 = 1200 So the answer will be between 800 and 1200

198

17.1 Developing written methods of multiplication

Continued 2

7

× 1

0

0

Calculate 270 × 4

4 8

0

2

Use the estimate to check that the answer is reasonable.

Answer: 270 × 4 = 1080 seeds

Exercise 17.1 Focus 1 Leanne collects 2 comics each month for a year. ACTION HEROES U N I T E

Tip In this exercise, remember to estimate before you calculate.

Tip



How many comics does she collect in a year?

Remember there are 12 months in a year.



199

17 Multiplication and division

2 Mr Singh has 13 boxes of tinned tomatoes. Each box contains 6 tins of tomatoes. How many tins of tomatoes has Mr Singh got altogether? Show your working.

3 Explain what is wrong in this calculation. Work out the correct answer. correct answer 4 ×

2

4

7 6

4

4 ×

7 6

2

4 Orla estimates the answer to 298 × 8 to be 2400. Is this a good estimate? Explain your answer. 5 Fatima reads 48 pages of a book. Parveen reads four times as many pages as Fatima. How many more pages did Parveen read than Fatima? Show your working.

200

Tip Start by working out how many pages Parveen reads.

17.1 Developing written methods of multiplication

Practice 6 Find the product of 58 and 5.

7 Multiply the numbers in two circles to give the number in the square between them. Fill in the missing numbers. a

b

19

17

133

7

3

4

8

8 Erik and Ollie complete the same multiplication. 100s 10s

Erik

3

4

× 1

2

8

1

2

1s

100s 10s

Ollie

5

3

4

×

0



Who has the correct answer?



What mistake has the other boy made?

1

4

1s 5 4

3

8

1

2

0

201

17 Multiplication and division

9 Milly buys 4 packets of red balloons.

Paula buys 2 packets of blue balloons. 8 blue balloons

16 red balloons

  

Milly says, ‘I have four times as many balloons as Paula.’ Is Milly correct? Explain your answer.

10 Here are some digit cards. 1

 

3

 

5

 

7

 

0

Use three of these cards to make this calculation correct.



×

= 150

Challenge 11 Use the digits 1, 2, 8 and 9 to make the multiplication that has the greatest product.



202

×

=

17.1 Developing written methods of multiplication

12 Find the mistake in this calculation. correct calculation 7

4

× 3



5

2

1

7

5

×

4

1 5

5

Explain what is wrong and write the correct calculation.

13 Here are four digit cards. 2



4

 

6

 

 

8

Use three of these cards to make this calculation correct. 0

×

=

3

0

14 Write the same digit in each box to make the calculation correct. 4

6

× 3



3

3

2

15 Magda says, ‘If you add three consecutive numbers, the sum is three times the middle number.’

Is she right?



Give three examples of 2-digit numbers to justify your answer.

203

17 Multiplication and division

17.2 Developing written methods of division Worked example 2 A group of friends earn $72 by washing cars. They share the money equally. They each earn $6. How many friends are in the group? A group of friends earn $72 by washing cars.

Read the problem carefully.

They share the money equally.

Underline the important information.

They each earn $6. You need to work out 72 ÷ 6

Decide which calculation to do.

Estimate: 60 ÷ 6 = 10

Estimate the answer.

120 ÷ 6 = 20 So the answer will be between 10 and 20 12 6 712 Answer: 72 ÷ 6 = 12 friends

dividend divisor quotient remainder

204

Calculate 72 ÷ 6 Use the estimate to check that the answer is reasonable.

17.2 Developing written methods of division

Exercise 17.2

Tip

Focus 1 Write the number that is half of 58.

Remember to estimate before you calculate.

2 Yuri has 96 triangular tiles.

He uses them to make hexagons like this.





How many hexagons can Yuri make?

3 Pencils cost 9 cents each. Myrtle has 52 cents.



9 cents each How many pencils can she buy? How much money will Myrtle have left?

Tip You need to find the remainder.

       4 Conrad has 50 eggs. A box holds 6 eggs. Conrad says he needs 8 boxes. Is Conrad correct? Explain your answer.

205

17 Multiplication and division

5 Apples are sold in trays of 4. Tara has 58 apples to pack in trays.

How many trays does Tara need to pack all her apples?

Practice 6 Petra wants to put 62 photos in an album. A full page holds 4 photos. She fills as many whole pages as possible. How many photos does she have left over?

7 Which pairs of numbers can be written in the boxes?



24 ÷

=

8 Put each calculation in the correct box.

25 ÷ 4 = 5

70 ÷ 7 = 10

76 ÷ 9 = 8



63 ÷ 7 = 9

84 ÷ 8 = 11

29 ÷ 3 = 9



76 ÷ 9 = 8

63 ÷ 9 = 7

45 ÷ 5 = 9

True

206

False

17.2 Developing written methods of division

9 There are 160 students in Year 4. A teacher orders 6 boxes of pens. Each box contains 24 pens. Has the teacher ordered enough pens to give one to each student? Explain your answer.

10 Find the missing digit. 2 4



3 2

Challenge 11 Which pairs of numbers could be written in the boxes?



48 ÷

=

12 Find the odd one out. Explain your answer.

48 ÷ 4   96 ÷ 8   84 ÷ 7   75 ÷ 5   72 ÷ 6

207

17 Multiplication and division

13 Use or = to complete these statements.

96 ÷ 4

96 ÷ 3



69 ÷ 3

96 ÷ 3



91 ÷ 7

84 ÷ 3

14 In the diagram, the number in each box is the product of the two numbers below it. Write the missing numbers. 96 8 2

15 Magda has two different types of tile.



rhombus

She uses 4 triangles and 2 rhombuses to make a ‘fish’.



208

triangle

Magda uses 56 triangles to make some fish. How many rhombuses does she use?

18 Position, direction and movement 18.1 Position and movement Worked example 1 What is the position of X on the grid? y-axis 6 5 4 3 2 1 0

0 1 2 3 4 5 6

x-axis

y-axis

You can use coordinates to describe a position on a grid.

6

First use the x-axis to find the horizontal position of the X.

5 4

The X is 3 squares across.

3 2 1 0

0 1 2 3 4 5 6

x-axis

209

18 Position, direction and movement

Continued y-axis

Next use the y-axis to find the vertical position of the X.

6

The X is 5 squares up.

5 4 3 2 1 0

0 1 2 3 4 5 6

x-axis

y-axis

Write the number as coordinates: (horizontal position, vertical position).

6 5 4

The point where the lines cross has coordinates (3, 5).

(3, 5)

3 2 1 0

0 1 2 3 4 5 6

x-axis

Answer: The position of X on the grid is (3, 5)

compass coordinates quadrant

210

18.1 Position and movement

Exercise 18.1 Focus 1 Complete the compass directions. N N

W

E

S S 2 Describe the direction of the path from flag to flag using compass directions. N

1 2 3 4 5 Start

Finish

211

18 Position, direction and movement

3 Draw an arrow from the coordinates to the cross in the correct position. Remember, the first number (x) is how far horizontally, the second number (y) is how far vertically. (↔, ↕) y-axis (2, 6)

6 5

(4, 4)

4

(1, 2)

3 2

(5, 3)

1 0

(3, 0) 0

1

2

3

4

5

x-axis

6

4 Mark the coordinates below on the grid. Join each coordinate to the next and then join the last coordinate to the first to make a polygon.

2 across and 3 up (2, 3)



4 across and 1 up (4, 1)



5 across and 2 up (5, 2)



3 across and 6 up (3, 6)



2 across and 5 up (2, 5)



What is the name of the polygon you have made?

y-axis 6 5 4 3 2 1 0

212

0

1

2

3

4

5

6

x-axis

18.1 Position and movement

Practice 5 This is a map of the town where Halim lives. Map of Halim’s Town key:

N W

E S

Halim’s home Bank Café Museum Park School Shop

0

200m

Travel agent

a What compass direction should Halim follow to get from his home to: i

the shop?

ii

the school?

iii the park? b What is the compass direction from the café to the bank? c What is the compass direction from the bank to the café?

213

18 Position, direction and movement

6 Write the letter that is at each of these coordinates. Rearrange the letters to reveal a word about the coordinates. y-axis 6 5

A B C D E F

4

G H

3

M N O P Q R

2

S T U V W X

1

Y Z

0

0

1

2

I

3

J

4

K L

5

6

x-axis

(2, 2) (4, 5)  (1, 5)  (2, 3) (3, 2) (6, 3) (1, 5) (5, 3)  

 

 

 

 

 

 

The word is:

7 a Mark the coordinates listed on the grid. A (3, 4)

B (0, 6)

C (3, 6)

D (0, 4)

y-axis 6 5 4 3 2 1 0

0

1

2

3

4

5

6

x-axis

Join the four coordinates in order with a ruler. Join the last coordinate to the first. b What polygon is made?

214

18.1 Position and movement

Challenge 8 Complete this compass with the compass directions, and write the number of degrees turn that each direction is from North. N 0 degrees NE 45 degrees

S 180 degrees 9 Label the coordinates of each

•.

y-axis ____, ____

6 5

____, ____

4 ____, ____

3 2

____, ____

1 0

____, ____ 0

1

2

3

4

5

6

x-axis

215

18 Position, direction and movement

10 Plot these coordinates on the grid. A (4, 0) B (0, 1) C (1, 5)

y-axis

6 5 4 3 2 1 0

216

x-axis 0

1

2

3

4

5

6



A, B and C are three vertices of a square. Complete the square.



What is the last vertex of the square?

18.2 Reflecting 2D shapes

18.2 Reflecting 2D shapes Worked example 2

mirror line  reflection

Reflect this shape in the mirror line on the grid. y-axis 7 6 5 4 3 2 1 0

0

1

2

3

4

5

6

7

8 10 11 12

x-axis

y-axis

This is a horizontal mirror line. C

7 6

One edge of the shape is along the mirror line.

D B

5

A A’s reflection

4 3

The vertices A and E are touching the mirror line. Their reflections will also touch the mirror line.

E E’s reflection

2 1 0

0

1

2

3

4

5

6

7

8 10 11 12

x-axis

217

18 Position, direction and movement

Continued y-axis C

7

2 B

5

2

A3 A’s reflection 1

4 3

3 E 1 E’s reflection

2

2

2

3 C’s reflection

1 0

D 1

1

6

0

1

2

3

The vertices C and D are both three squares from the mirror line. Their reflections will also be three squares from the mirror line, on the other side of the mirror.

4

5

3 D’s reflection 6

7

8 10 11 12

x-axis

y-axis C

7

2

2 B

5

A3 A’s reflection 1

4 3

1

3 E

1

1 E’s reflection

2

2

1

0

2

3

4

Join the vertices to make the reflection of the whole shape.

2

3 C’s reflection

1 0

D 1

1

6

The vertex B is one square from the mirror line. Its reflection will be one square from the mirror line, on the other side of the mirror.

3 D’s reflection

5 6 7 8 10 11 12 B’s reflection

x-axis

Answer: y-axis C

7 6

2 B

5

A3 A’s reflection 1

4 3

2

2

3 C’s reflection

1 0

218

D 1

1

0

1

2

3

4

2 1 1

3 E 1 E’s reflection 2 3 D’s reflection

5 6 7 8 10 11 12 B’s reflection

x-axis

18.2 Reflecting 2D shapes

Exercise 18.2 Focus 1 Complete the reflection of these shapes in the mirror line. y-axis 11 10 9 8 7 6 5 4 3 2 1 0

0 1 2 3 4 5 6 7 8 9 10 11

x-axis

2 Complete the reflection of these shapes in the mirror line. y-axis 11 10 9 8 7 6 5 4 3 2 1 0

0 1 2 3 4 5 6 7 8 9 10 11

x-axis

219

18 Position, direction and movement

3 y-axis 6 5 4 3 2 1 0

0

1

2

3

4

5

6

x-axis

a What are the coordinates of the vertices of the triangle?

(

,

)   (

,

)   (

,

)

Tip Remember (↔, ↕).

b Reflect the triangle in the mirror line by counting the squares. c Write the coordinates of the reflected triangle.

(

,

)   (

,

)   (

,

)

d What shape is made by the original triangle and the reflected triangle together? Practice 4 Reflect these shapes in the mirror line. y-axis 11 10 9 8 7 6 5 4 3 2 1 0

220

0 1 2 3 4 5 6 7 8 9 1011

x-axis

18.2 Reflecting 2D shapes

5 Reflect these shapes in the mirror line. y-axis 11 10 9 8 7 6 5 4 3 2 1 0

0 1 2 3 4 5 6 7 8 9 10 11

x-axis

6 a List the coordinates of the square on the grid.

( ( ( (

y-axis



6 5 4 3 2

, , , ,

) ) ) )

1 0

0

1

2

3

4

5

6

x-axis

b What shape will be made by combining the square with its reflection in the mirror line?

c Draw the reflection of the square in the mirror line to check your answer to question (b).

221

18 Position, direction and movement

7 Draw a rectangle on the grid that will make a square when combined with its reflection in the mirror line. y-axis

6 5 4 3 2 1 0

0

1

2

3

4

5

6

x-axis

Challenge 8 Reflect these shapes in the mirror line. Count half a square where the edge of the shape is halfway between the grid lines. y-axis 11 10 9 8 7 6 5 4 3 2 1 0

222

0

1

2

3

4

5

6

7

8

9 10 11

x-axis

18.2 Reflecting 2D shapes

9 Reflect these shapes in the mirror line. y-axis 10 9 8 7 6 5 4 3 2 1 0

0 1 2 3 4 5 6 7 8 9 10 11

x-axis

10 Answer questions (a) and (b) before you draw the reflection of the shape. y-axis 6 5 4 3 2 1 0

0

1

2

3

4

5

6

x-axis

a What shape will be made by combining this pentagon with its shape reflected in the mirror line?

b List the coordinates of the vertices of the reflected shape.

c Draw the reflection of the shape on the grid and check your answers for (a) and (b).

223

18 Position, direction and movement

11 Draw a pentagon on the grid that will make a hexagon when combined with its reflection in the mirror line. y-axis

6 5 4 3 2 1 0

224

0

1

2

3

4

5

6

x-axis